JHEP01(2020)007 ) and a B Springer , a closed g e generalized March 27, 2019 January 2, 2020 : December 9, 2019 : Gravity, Super- November 15, 2019 : tring effective ac- : with the help of a Received TM Published Revised Accepted ppropriate stringy symme- . Further results include a aded Poisson structures and ically involves a doubling of rticular natural deformations e point of view of graded ge- nd torsion on the generalized ndles. A byproduct is a unique et and new formulas for torsion cies of traditional approaches to ly associated to a non-symmetric Published for SISSA by https://doi.org/10.1007/JHEP01(2020)007 that are based on a metric M . Projecting onto [email protected] [1] , M T ∗ T [2] ∗ ⊕ T , we obtain a connection and curvature invariants that TM E ∼ = E (locally expressed in terms of a Kalb-Ramond 2-form H . 3 1903.09112 . The derived bracket formalism relates this structure to th The Authors. φ Differential and Algebraic Geometry, Classical Theories of c

In recent years, a close connection between supergravity, s , [email protected] E-mail: Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany gravity Models, Superstrings and Heterotic Strings ArXiv ePrint: Open Access Article funded by SCOAP Abstract: Eugenia Boffo and Peter Schupp Keywords: and curvature tensors associated toKoszul-type generalized formula for tangent the bu torsionfulmetric, connection which natural resolves ambiguity problemsnon-symmetric and inconsisten gravity theories. reproduce the NS-NS sectorfully of generalized Dorfman supergravity bracket, in a 10 generalized dimensions Lie brack Deformed graded Poisson structures,geometry generalized and supergravity ometry, introducing an approach basedderive on the deformations corresponding of gravity gr of actions. the We 2-graded consider symplectic in pa manifold tions and generalized geometrygeometric has structures. been discovered We that investigate typ this relation from th tries, and yields a“doubled” connection tangent with bundle non-trivial curvature a differential geometry of a Courant algebroid, which has the a natural non-isotropic splitting of scalar dilaton Neveu-Schwarz 3-form JHEP01(2020)007 8 1 4 8 22 24 25 26 27 28 10 12 16 17 19 20 30 33 12 20 = 0 } ν ibe interactions. This ap- , x φ µ x ism and is an arguably slightly { stance the inclusion of magnetic d deformations of canonical com- , µ ν δ = } and dilaton ν , p M B µ x – 1 – [1] { T , [2] ) ∗ x ( T , 2-form ]) and it can also deal rather elegantly with first order g µν 2 eF = -model } ν σ , p µ p ] (see also [ { 1 5.3.1 Koszul5.3.2 formula Metric connection with torsion 5.2.1 Courant 5.4 Supergravity bosonic NS-NS sector 5.1 Deformed graded5.2 Poisson algebra Derived structure 5.3 Connection and curvature 4.1 Generalized torsion,4.2 Lie bracket and Connection connection for4.3 the derived deformed Comparison Courant with4.4 algebroid other definitions of Curvature torsion invariants 3.1 Deformed graded3.2 Poisson brackets Derived structure actions. In the electromagnetic case, the deformation proach is well-established inmore the general context alternative to of gauge electromagnet monopole theories. sources It [ allows for in 1 Introduction Deformations of Poisson structures inmutation classical relations physics an in quantum mechanics can be used to descr A Generalized differential geometry 6 Discussion and comments 5 Deformation with metric 3 Graded Poisson algebra of 4 Courant algebroid connection Contents 1 Introduction 2 Summary of essential notions JHEP01(2020)007 B , a , g g in terms of has been ex- . Locally, on tal derivative. φ dB M ∗ e correct Lorentz = T dinates. But unlike sed on deformations ⊕ H have been classified by fields, turns out to be correspond to canonical M more familiar Lie-bracket, TM rgravity. The deformation λ ). This is a simple appli- ations are thus necessarily also the coordinates of the ∗ ubled (tangent plus cotan- of phase space coordinates suitable integrability prop- higher geometric analog of µ s General Relativity on this ed as x on suitable projection onto oned above, the fields pergravity, as the supersym- s a Riemannian metric eformed “free” Hamiltonian T y on the generalized tangent ( ioned Generalized Geometry ∂ t pairing on sections, an an- ts of Riemannian, symplectic m this point of view. Here we µ ⊕ tive field theory of superstrings ns act non-trivially — as is in = alized Geometry. The deforma- annot entirely be formulated in ). A ed structure. this observation shall not be the ibility conditions between these R µ TM ), or extensions of it, interpreted as δA M, ] and others show this in the frame- ( ∼ = 7 ]. 3 – 8 d, d E ( 5 H and a dilaton scalar field O H – 2 – . Globally, the deformation is non-trivial whenever the /m ν p , which is defined up to 1-form Λ (gauge) transformations )) generated by a gauge field ]. Gauge transformations = B x with structure group 3 ( ν µ x M ∗ eA T + with ˙ ⊕ µ ν ˙ x , p µ µν TM x in conjuction with the deformed Poisson structure yields th ( Λ. The 1-form gauge parameter Λ is itself defined only up to a to eF d m ] in terms of the third cohomology class 2 7→ = 4 + / generate a deformation via a local change of phase space coor ) µ 2 µ p B p φ In the recent past the generalized differential geometry of As mentioned above, here we shall take a different approach ba So far little is known about gravitational interactions fro The appropriate geometric setting needed to accomodate all , p = 7→ µ x ˇ doubled vector bundle. For example, the works [ ploited to show — with variousmetric suitable theory assumptions of — gravity that in su itsof own type right, IIA but also and as IIB, the can effec be described as some kind of Einstein’ cation of Moser’s lemma [ a contractible patch, the Neveu-Schwarz field can be express work of Generalized Geometry. Inbase Double manifold Field (spacetime) Theory, where arestandard target doubled, spacetime similar — results were found, — see up e.g. [ closed Neveu-Schwarz 3-form field strength ( of the canonical Poisson structure is based on a local change and will show how to implementdata the that approach in will the enable context us of supe to formulate a Supergravity action i transformations of the deformedH Poisson structure. The und the electromagnetic case, the deformedterms Poisson of structure gauge c invariantfact quantities, to since be Λ expectedlocal, transformatio in but view they of nevertheless the extend gerbe to structure. a globally The well-defin deform Globally, this is thea structure line of bundle). an In abelian analogy bundle to gerbe the (a electromagnetic case menti the Kalb-Ramond 2-form force ˙ objects must also hold. Exact Courant algebroids a generalized tangent bundle.bundle The natural is notion encoded of in symmetr chor Courant map algebroids into tangent with space an andthe ad-invarian a Dorfman Dorfman bracket bracket. is Unlikeerties not the and anti-symmetric it in satisfies order a to restricted assure Jacobi identity. Compat B field strength belongs to a non-trivial cohomology class. Severa [ of graded Poisson structures. It is known that the aforement graded Poisson structures and ittion is data closely in related fact to definesmain Gener a so-called focus generalized of metric, this but paper.and Generalized complex Geometry geometry. unifies aspec Itgent) typically bundle involves the study of a do JHEP01(2020)007 , ] e 4 M M ∗ ] for T [1] 17 T ), even- ⊕ x ]. Worth ( [2] -manifolds ∗ of this pa- φ 13 TM T ] and [ NQ et. We will see to the mapping 16 ectic manifolds by ˇ Severa’s letter 7 [ the bracket and the M,N in lativity nature” of super- ], further analysed graded olds. The notation will be of the AKSZ construction TM al natural geometric struc- Hamiltonian as differential. ]. Graded Poisson algebras s and other applications to n symplectic Another closely related and 10 metric setting with dilaton. vity from a derived general- ra encoded in local vielbeins, ⊕ e BRST and BV quantization anifolds 11 lynomial functions, was related st kind are given and a general e construction uses the derived ions of torsion in the context of derived bracket formalism. The ized connections and associated ymplectic manifold is devoted to a survey of differ- M trings. An interesting side-result and a scalar (dilaton) ions for a generalized Lie bracket, he derived bracket formalism: the f graded manifolds are the AKSZ ∗ 4 we will review the necessary back- T B 2 ≃ M [1] ∗ , a 2-form T , all these methods are applied with a concrete g – 3 – 5 ⊕ , admitting a Hamiltonian (shifted) vector field, ], and Cattaneo and Felder in [ M 15 M ] and in his PhD thesis [ [1] -model. [1] 9 σ T T [2] ∗ the general deformed, i.e. non-canonical Poisson structur T In the following section 3 ) is analysed and its Courant algebroid counterpart on on ). For further details see e.g. the review on supergeometry [ M -dimensional target spacetime. Indeed, this is the program that relates these objects to the generalized Dorfman brack d [1] T related to the Poisson ], Schaller and Strobl in [ 4.4 M,N 0 ]. They associate topological field theories to graded sympl 14 [2] ∗ N being 12 T ( [1] M T ∞ It is natural to conjecture that the “generalized General Re C = tually yielding the effective Lagrangianis for type a II generalization closed of s the Koszul formula to a non-symmetric space Map( Structure of the article. per: starting from awe deformation reconstruct of the the graded NS-NSized Poisson sector connection algeb of and corresponding 10-dimensionalbracket curvature supergra invariant. approach Th to Generalizedtures, Geometry which and we some shall addition describe in detail. gravity can be described in terms of the differential graded s section includes a comparison withGeneralized previously Geometry. proposed Finally, not in section deformation based on a Riemannian metric ential geometry in this settingtensors. and This in section particular contains several totorsion new general and results. curvature Definit tensors, connectionproposition symbols of the fir how a connection with non-trivial curvature arises from the ground on Generalized Geometryfixed and in graded this symplectic part. manif on In section by Ikeda [ N fruitful setup for the exploitationmodels of [ the rich structureslifting o the graded and symplectic construction on a pair ofmentioning, m amongst the numerous works, is the exploration and graded Lie algebras areof also the relevant in path the integral context of of th field theories with local symmetries. to Alan Weinstein. Roytenberg,symplectic in [ manifolds and found a 1-1 correspondence betwee with its sheaf of graded Poissonto algebras generated the by exact the po Courant algebroid on 2-graded symplectic manifold structures are special cases of graded Poisson algebras in t of degree 2pairing and are Courant derived algebroids. bracketsThe of relation The the between Poisson key CourantLie bracket was algebroid with algebroids to and the are notice derived explained that bracket by Kosmann-Schwarzbach in [ is computed via the derived bracket approach. Section with JHEP01(2020)007 ), is = E E ], to E (2.2) (2.1) (2.3) (2.4) → → 19 M anchor ∗ TM , together : ·i T , s is an injective ⊕ h· j TM : (called the by the bundle be a homomorphism j ) consisting of a vector . ) ·i ⊕ , TM s M h· ( , being the natural projection → ] · a lot of attention in the early ] for the second entry only: pts of Generalized Geometry. ∞ . , sotropic splitting to project a ρ is an embedding · be shown to follow from these ), , · , i i E C [ · , h E ˆ ˆ r form on sections e e ence holds (i.e. if : 0 ∈ ˜ ˜ e, e, Γ( ρ h h f vanishes on its image — in order to ) ) → ∈ E, ρ, ∀ e e ( ( ′′ ·i ≡ ρ ρ ] and by his student Gualtieri [ , , , e ] h· ′ TM ′ 18 = = is not unique and it does not need to be M ∗ i i ρ ] ] −→ s e, e ) e, e ˆ ˜ T e e [ ρ ∀ E f ⊕ e, e, [ – 4 – ) and a map j + −→ ˜ e, ′ E and the last two axioms can be polarized (using h e ] + [ˆ TM = Ker ]] , ˆ e ( Γ( ) M + ′′ ∗ j f e, i TM ) [˜ T | ˆ e → e e, e , that is the Whitney sum of the tangent space and the ( )] [ e, ] ) ′ ρ , h ˜ → e e ′ E ( e 0 M . e, , [ Γ( , ρ i i h ′ ′ ) [1] ] = ( ′ e ] + [ × T ( , e , e ′′ ′ ′ ]) fulfills the first axiom and the Leibniz rule and its bracket ) ρ · e e ⊕ , , e h h E · ] e, fe ) ) ′ [ [ e e M ( ( Courant algebroid ρ ρ e, e ]) = [ [1] ′ 2 2 1 1 ] : Γ( · ∗ E, ρ, A is surjective and Im , ( = id. It is then straightforward to show that · T = = e, e ρ i i s ]] = [[ ([ ′ ] ≃ ′ ′′ ρ ◦ , e ] , e , e ρ ′ ′ ): ′ M e e e ∗ [ [ e, e T + ˆ [ e, e, h h e ⊕ The canonical example of an exact Courant algebroid is given 1. [ 3. 2. Lie algebroid := ˜ ′ In this form the last axiom also yields the Leibniz rule of [ with a bracket [ bundle with a point-wise non-degenerate symmetric bilinea such that: three conditions. Forof example brackets, the anchor map can be proven to This is a minimal set of axioms; other possible relations can Definition 2.1. which we refer for aThe nice key explanation underlying of structure the is underlying that conce of a Courant algebroid: e The original works on Generalized2000’s Geometry, which are received usually considered those by Hitchin [ 2 Summary of essential notions A skew-symmetric. Courant algebroids for which the following short exact sequ inclusion map, are called exact. A splitting of the exact Courant algebroid TM cotangent space together with their natural pairing and wit such that establish the isomorphism. Wegeneralized shall connection in to fact the later tangent use bundle. a non-i isotropic — in the sense that the bilinear pairing is a vector space isomorphism. The splitting JHEP01(2020)007 , , M M and (2.7) (2.5) (2.6) [1] [1] . Via tional ormed T T iso TM s are given ⊕ [2] ∗ vanishes on (which also [1] T M ∗ ·i T [1] , M , which we shall most commonly h· Dorfman bracket T M he aforementioned on [2] that is defined with ∗ } ρ i T denote a pair of local = 2. From the graded x opic splitting slightly different — the ] in terms of the third { ζ | of ): All coordinate-dependent i 4 cture and properties. For ∗ p + | iator.) ρ natural pairing of M tisymmetrization, called the , ( ). . i ). Then the f preferred frame (coordinate Y 3 olds despite the shifts in the f a manifold Ω dη , ) are coordinates on the fibers ]. We will focus on M aded coordinate system that we = = 1 i inates on ) is part of the local Darboux ∗ , dx Y , i H. ∈ p i ˇ | Severa [ ι r frame, which however turns out 20 i θ T ∂ Y α V ι ξ − H dθ | , Γ( X ζ η , ι ∧ ∈ and X i has degree 2 and is given by + + i L is the adjoint = 0 χ ). Any two isotropic splittings are related dχ X η, ζ M ) with ( | j i i ] + Dorf + ] x 2.6 [1] = | i , θ i T (i.e. odd coordinates on the fibers of . Hence, exact Courant algebroids are classified – 5 – χ dp U ) and d X,Y [2] U, V α ∧ ∗ ξ dB i T = [ Let := [ TM + and the momenta ( dx are 2 . H Γ( ] H
= i M Dorf ∈ ] ω , θ 7→ i U, V [ ˇ χ Severa’s classification this will clearly require some addi H are assigned as U, V ). Let us briefly recall the construction without going into [ X,Y Z R is isotropic in the sense that the bilinear form ) := ∗ α M, , i.e. ρ ξ ( |·| ∈ 3 M ). We shall later extract a generalized connection from a def ∗ H R T (Dorfman bracket) ⊕ M, ( 3 ), where ( i TM H . Weights is given by: . The non-skew-symmetric bracket for the Courant algebroid , p the exact Courant algebroid is then seen to be isomorphic to t α ∈ M ∗ -transform that maps ] ρ , ξ Dorf TM [2] i . The inclusion ] B ∗ Graded symplectic manifolds are manifolds very rich in stru The canonical symplectic form of Exact Courant algebroids have been classified by The Dorfman bracket can be twisted by a 3-form H x ⊕ ) are coordinates on the base T M i ∗ iso x U, V the basics on graded manifolds and algebras, we refer to [ geometric structure like a particularto choice be of quite splitting natural. o We will comment on this in due course. Courant algebroid. In view of canonical example with the twisted bracket ( s by a cohomology class the shifted cotangent bundle of the shifted tangent bundle o onto its image. It can be shown that one can also always find an isotr basis and its dual) and we can identify ( coordinate choice, from a bundle point of view it is a choice o by [ expressions that follow willhave described refer here. to the local canonical gr interpret as (target) spacetime.grading. Darboux’s In theorem an still affine h Darboux chart corresponding to a chart T respect to the bilinear form of the Courant algebroid and the used in Generalized Geometry isCourant the bracket, Dorfman will bracket. not (ItsJacobi an be identity used in the here, definition as would the holdDefinition axioms only 2.2 up would to be a Jacob by ( sections of details: a natural choice for the inclusion map coincides with a local trivialization of the bundle), coord where the bracket on vector fields is the standard Lie bracket [ ( geometry point of view the choice of of JHEP01(2020)007 , ry nd dξ (2.9) (2.8) = (2.11) (2.12) (2.13) (2.10) ω ) with a Q ι n )-invariant ( − d, d ( O ) will play the role . x = 0, it is a symplec- ( ·} , ω αi st satisfy the structure ρ Q Θ , . 2: inetic term linear in the Hamiltonian ) and compatibility with { L − x with the . , ( = fined by belonging to the degree +1 , ] = 0 = 0 ω : 2 αi graded setup. ads to the fundamental result γ αβ Q } } , i.e. Q ρ ξ η j gree j , e ee 2 and the Courant algebroid f , ω β 1 ) ξ geometrical and non-geometrical ’s, is a completely antisymmetric , p , θ e = e i i α ξ ( that can be obtained as a derived ξ , on θ p } ρ = 0. { { ) = [ β C 2 x = } , ξ ( 2.1 2 Q , , α } . ξ , e { αβγ , e , } = 2 } = 0 = 0 1 i C = 0 2 } } + 1 , e , f 3! j } } α is a vector field , e Θ Θ 1 , denoted by Θ: Θ – 6 – , ξ + , χ being the degree of e , i n i = 0, as can be seen from the Jacobi identity and i {{ {{ h p χ Θ p n Q, Q 2 { must therefore be of degree 3. The most general ) { = = = { { Q x } } } ( 2 2 , M , j αi 6= 0, j , f i i , e , e ρ . The position-dependent matrix δ } δ [1] 1 } α e T Θ is symplectic homological iff it is ξ = abc Θ = ) together with the graded Jacobi identity for the Poisson + 1 { , e, } [2] } H 1 n Q j j ∗ e {{ , due to the odd parity of the 2.10 T Θ = , θ ), such that , x {{ i (the set of polynomials of degree 3). It can be interpreted ve i C and ), we shortly recall the notion of a homological vector field a E p χ c 3 ·i { { , =: ab Π( O h· f γ X is hence recovered from the Poisson bracket (of degree homological vector field , , ξ ] · β bc , A ξ Q ): a · [ α Q ξ 1, 2.3 , ) x − ( abc M, ρ, ∗ R 6= . αβγ T η n some exact form) and C ⊕ The master equation ( Before reviewing the derived brackets construction that le If A Hamiltonian Θ for When such a vector field preserves the symplectic form dξ TM of the anchor map ofor a master equation Courant (corresponding algebroid. to The Hamiltonian Θ mu the graded Poisson structure must also be checked. the definition naturally as the tensor representing“fluxes”: all the T-dual stringy This implies various constraints on the rank-3 tensor function in the sheaf structure. The bracket, the anchor map and the pairing are de bracket implies the axioms of a Courant algebroid where while the corresponding canonical Poisson brackets have de pairing corresponding Hamiltonian of weight 1 + The last line can be written more compactly as degree 3 Hamiltonian Θmomenta that and can of be a written potential down cubic consists in of the a degree-1 k coordinates under which conditions it gives rise to aDefinition Hamiltonian 2.3. in this on the relation between this( dg-symplectic manifold of degr shifted vector fields for tic homological vector field. JHEP01(2020)007 . )- } 3 ) to d, d , ξ }} imply (2.17) (2.14) (2.15) (2.16) ( } 3 2 ressions O , , , ξ M , ξ } } 1 E,TM ). [1] 1 , ξ T , η , ξ Θ } one can recon- ). [2] axiom 2 axiom 3 must satisfy the Hom( ∗ Θ {{ Dorf R ] , , · M T T ∈ bniz rule and the i } , i
Θ · 2 ] 2 ) ρ [ M, [1] 2 ) and the constraints ( x , ξ , ξ T 3 ( ] , ξ i Θ 2 2 {{{{ a 2.9 H [2] ξ ). From now on we will 1 2 [ ∗ , ξ ). M, ρ, {{ } , , ρ . These functions can be 1 ical case, whose symplectic ∗ 3 ) T 1 M ξ ) on the graded symplectic + [ ξ T ∗ x e tangent space. Finally, the 2.1 ]]+ e proof of the correspondence , ξ M h h ( ection. Moreover the anchor id can be given as follow. The 3 } or the Poisson graded algebra: } T ⊕ 3 ai 2 2.9 [1] , ξ ρ ⊕ 1 , ξ , ξ = 2 = 2 T ξ } } [ } 1 ) and the axiom is verified. TM , [2] , Θ 2 }} 2 ∗ , 2 , , ξ ) = TM ξ , ξ } T } x 2 } , ξ 2.10 ( 2 Θ } ) = 0 , ξ , 2 αi ] + [ x , ξ } ) = 0 3 ρ ) is precisely the Dorfman bracket given in ( Θ } , ξ x Θ 1 bi , as defined in ( ( , ξ , Θ ] i ρ ] 1 2 , ξ ˇ ) Severa and Weinstein classification of Courant 3-form introduced previously as the component 2.11 M ξ α {{ – 7 – | {{{{ x ∗ Θ , ξ , ρ ( 2 1 1 1 H T j j ξ ξ a ∂ {{{ { {{{{ + ⊕ ρ k | ) immediately tells us that the pairing is the b β = [[ = [ a = 2 = 2 ρ δ TM }} 2.13 ( 3 }} 2 , ξ } , ξ 2 . According to 2 ξ , ξ { M , ∗ Θ } T 1 {{ ’s with a basis of sections, and all following coordinate exp = 0. These relations can be solved by choosing is given, the data and the algebraic relations for , α , ξ } . Furthermore, the anchor } ). Equation ( ξ 1 η Θ being functions of degree 1 on M Θ , , ξ 2.7 ) is understood as the same map appearing in ( 1 {{ [1] Θ Θ T { is the representative in the third cohomology class = ), possibly twisted with the {{ i ∈ O [2] 2 H ∗ 2.5 2 E,TM , ξ T ]]= 2 3 , e ξ 1 h , ξ ) 2 that lives only in 1 ξ Hom( Hence from the canonical graded symplectic structure of For instance, let us apply these considerations to the canon For what concerns the remaining two axioms, the (graded) Lei e, e [ ξ , C ( ∈ 1 ρ ξ [ as found from following differential and algebraic equations be constant and nullderived on bracket forms, for i.e. the it Courant is algebroid the ( projector onto th always identify the struct the canonical example of a Courant algebroid, ( will refer to theρ local coordinate basis introduced in this s identified with elements of the space of sections Γ( those for a Courant algebroid on equation ( form is given in ( manifold between the dg-symplectic manifold andfirst the Courant axiom algebro is a consequence of the (graded) Jacobi identity f due to the master equation impose some restrictions on it. Th The last term is zero due to the master equation ( invariant constant algebroids, of for (graded) Jacobi identity give directly: Hence, we have shown that when a Hamiltonian Θ in ( JHEP01(2020)007 , ), ∋ 2). ) =0 M (3.1) (3.2) } M l [1] and R: β s d is the iated to , p [2] T i is metric ∗ dξ ∇ p ⊕ T , { ction. The ∧ ( ∇ G i ). βl ∞ M ρ αβ C M ), as implied by [1] αi
( ∗ 1 . ρ ) M ∞ β − T ty and the graded ( ξ ], the action of the C G 2 [2] α 9 preserving the metric. ∗ ξ v, w S ∈ egree 4 equation for a T iα + ) γ es from a Lie algebroid e that will be clear from ( the momenta and those ∈ M } x , because those quadratic each involving a different 2 Γ ( and R resulting in expres- ϕ p ) just outlined. The most ion, where is connection γ + R( [1] , Λ ξ f ξ αβ ) ǫ neralization of the canonical T h ∇ ξ g and the anticommutativity ∈ G Lie i δ , ] r in ⊕ , ), which we require to preserve M, ω ions are placed on ) ij and G , ξ U, U, V dx f v γ M ( [1] 2.8 α ( ξ v, w − ξ G v ∇ T β [1] ) j ξ ∗ x = = [ = = α [2] ( T ξ dx ∗ α ). They are listed in the left column in } } } } { T ∧ U ) M on 3.1 i x v, f v, w δ ( v, U U, V { := [1] ξ { { { . i δǫϕ jγ T δ – 8 – C Γ dp ) V,U [2] , x ∗ ∧ γ αγ ( ∋ i ξ ]. The Poisson bracket of two momenta is given by
T , ) 1 9 α β ξ dx − αβγ ξ M ) , G ) C ) + x ( x x [1] β ( ( 1 ij T iα β 3!3! , iα dξ αβ β αγ j Γ ⊕ i + ∧ β G Γ δ ) places further restrictions on ξ o i , the commutativity of the bracket and the resulting degree ( M δ = = = R = ξ + ξ invertible symmetric block matrix, αβ } } } } γ [1] ). In fact, according to Roytenberg’s work [ , which corresponds to the inverse of the Poisson bivector, i
β 2.10 j j ξ d ∗ β α 1 x ξ ω 2 β ( T − α , p , ξ . In formal terms, the algebra of sections of the Lie algebroi , x , ξ ξ ( i ξ i i can be understood as a splitting of the Atiyah sequence assoc ) α G × αβ p G p p ij ξ x ∞ covariant differential operators { { { ( d G { αβ h ∇ C ’s is R α , ξ βγδ h i -manifolds of degree 2 [ i dξ p ) we must expect a degree 1 object with the properties of a conne N ,C x + i dx ( ξ p i bracket and (of course) still obeys the graded Jacobi identi = v n p )-invariant pairing between vector fields and forms in a sens ω αi - actually act trivially) on the functions of lower degree com := ρ x Similarly, from the bracket between the functions linear in α ξ should be a 2 d, d ξ ( 2 3! Jacobi identity with another degree 1 object implies that th with respect to linear in The symplectic form This is a non-degenerate Poisson structure if some restrict as the identities ofof the the bracket Poisson imply algebra, directly.O the It degree should countin be considered as a ge the particular deformation one wants to analyze. w, v G Atiyah algebra of action preserving The connection the curvature 2-form of the aforementioned connection, R polynomial functions of degree 2 (which are just those linea components and in the right column in a coordinate free notat the 3.1 Deformed gradedConsider Poisson again brackets the graded symplectic manifold ( Leibniz rule is given by the following brackets ( the Jacobi identity with sions that can bemonomial decomposed in into the a setquadruplet degree of of 1 degree and 4 2 equations, generators. In particular the d 3 Graded Poisson algebra of in general deformation of the canonical Poisson structure ( The master equation ( symplectic JHEP01(2020)007 ] e η ⊕ ) 22 that M = 0, ρ d, d [1] , bi ( ∗ ]. From ρ O T , and the 21 = 0 l ∇ p o , ich the β l τ ero unless stated ρ ξ β γ and itself gives the ξ ) , ξ gent bundle i k x eful in the derivation of p ( p i is thus forced to be the = 0 (3.3) il β n block. ρ jects built from a pair of geometric fluxes and their k  γτ s to discuss. In this paper γ d tion of the dilaton. In the ∇ one can consult [ t is null on forms, egree-preserving morphisms ρ R βl αβ erived” connection with non- × ρ
quivalently a generalized local βl eady been used before in [ k 1 αγ M d ρ ated too. The option for ∂
ck connection can also be traced − in the fluxes was done in a local 1 on regarding global implications, must have degree 2. This can be αi [1] r sense) metric connection said to G + ρ − T h R β G δl ξ α ρ ξ [2] α : ∗ ξ α p kδ . Therefore, without loss in generality, ξ β T − ρ 2 Γ  , ) = αk x ( – 9 – = 0 C j from the ordinary affine connection , ρ x is underdetermined. An ansatz must be made in p i ) ρ p ρ 2, the generators . This equation can be solved for R as a function of x has a null lower right j = 0 ( δ − G ρ j G C δ Let us now discuss briefly a class of canonical transforma- and ρ requiring a global vielbein or parallelizable manifold. Th βδ 1
and of the anchor 3!3! γδ Γ 1 ’s together with one
− ξ , not 1 G ) and focus on the fluxes to keep track of the modifications G − G ) + G x i 3.1 ( γ ρ γ C i ) ξ ρ αγ (
ρ 1 ∇ − 2 3! G } , of the metric is the differential graded connection on the generalized tan β C too has an interpretation in term of E, as that one metric to wh , ξ () α ρ induced via the anchor map ξ ] in the related context of Double Field Theory, and proven us G ∇ { 23 The master equation for the bracket between M [1] Canonical transformations. tions for this algebra,and the since inner the automorphisms. brackets have These degree are d is reduced by the vielbein in a local neighborhood. which is also equivalent to: This set of constraints on remaining couple ofmomenta relations. and from These a pair involve of the degree 4 ob metric coordinate chart and in the presentback context, to the fixing Weitzenb¨o a localvielbein frame E. for the We degree-1 are coordinates, certainly or e Supergravity theories. In orderlet to avoid us potential mention confusi that the above discussion of absorbing otherwise. (Later in thezero paper we curvature.) will consider In a the different absence “d of curvature the connection and [ induced by the curvature.Bianchi For identities a in complete the analysisnow canonical on on dg-symplectic the the non- Poisson bracket of the momenta and hence R is set to z we pick up here, also implies that curvature-free but generically torsionful (inbe some of broade type. Weitzenb¨ock connections Weitzenb¨ock have alr discussion we will hint to other choices that can be investig i.e. a projection onto tangent space up to rescaling by a func where which gives order to solve it,we depending shall on choose the a deformation (generically that one non-constant) want anchor map tha T interior product is performed with we can take R = 0 in ( the fluxes JHEP01(2020)007 . (3.4) (3.5) (3.6) } A. ] 2 σα ), and the via Poisson ,h , as defined G K, δW 1 ρ n h { [ νβ , δ The associated 2.10 O ); +  G 1 ’s generates local = } M ξ σα ). ( } γνσ [1]). One recognizes G ∞ ebra of functions on C ∗ 3.3 ,A ket construction, and γ νβ C T 3 ξ δK, W G h ∈ in the { { f ) + lgebroids). The algebra of rary degree γνσ = End( = β we have: of the Poisson bracket. Con- C ; } we have already discussed the ∈ and anchor map mmetry of the gerbe structure 2 ation. , ξ i h is still linear in the momenta }} e a modified Poisson structure e the Courant algebroid pairing . α − + β 2 with ·i A ,A ξ n β like: , 2 } , ξ . ξ δβ f ∇ tonian. Fluxes can be easily added later, 2 h· α ) n α ( h, W G ξ α ξ { , h h ) ξ ∈ O G ere mostly for notational simplicity. 1 kα ( x δ ∈ O h = ρ ( K, + Z Γ { [1]) and , written out in components, is given by {{ α ′ k αβ = ∗ ǫ = ] αβ ξ · ) has the following structure: + ρ } ) in section T β , = } G M ). , pairing ξ ( · } ′ α ·i γǫ ,A ] 2 2 1 · , } = , ξ }} G 2.13 , X 3.1 h· · } + ,W } – 10 – − ∇ , + ∈ i } β K,W ,A β , f h, A p 1 { i β ξ , ξ { ), satisfying the master equation ( h kα ) Θ Dorf α γ α ] { ξ x · ξ h, K = ) and ( , ( Γ {{ 2.9 , { 2 · ∇ k v ), whose master equation implies ( [ with β [1]), {{ closes as assured by degree counting and by the Poisson h : ρ δA = T n = 2.12 h 2.9 = ( αβ − ). } 2 O h β G ·i Λ ), ( M, ρ, }} , kβ of ∗ }} − { , ξ γ a function with a term linear in momenta and one quadratic in ∈ h· } T Γ , α h 2.11 ,A k B α ⊕ 2 K,W , ξ ρ h ’s, see the relations ( { Dorf K,W  ] { Θ ξ · , γ , h, TM 1 · ξ {{ { [ h { = = = ′ ] = } β M, ρ, with Hamiltonian Θ ( A ∗ , ξ ] h, Z α T 2 ξ { M [ , δ ⊕ 1 = [1] δ [ ) transformations: T As an example for the correspondence between the Poisson alg These canonical transformations are of course derivations TM Later in this section we will ignore the flux terms in the Hamil δZ 1. The pairing is given by 2. The anchor map follows from 3. The fully generalized Dorfman bracket [ -transformations are actually automorphisms for Courant a [2] 1 d, d ∗ ( B brackets themselves, since they produceand a new quadratic function in whic the bracket, the infinitesimal canonical transformations look The algebra of these generators When acting on an element of the algebra of functions of arbit Courant algebroid ( canonical graded Poisson structure.together Now with we the will Hamiltonian ( investigat Courant algebroid with Dorfman bracket [ T 3.2 Derived structure In this part wedescribe want the Courant to algebroid focus corresponding on to the the deform outcome of the derived brac they are essentially along for the ride and we suppress them h achieved by choosing for The momenta generate diffeomorphisms and the term quadratic the degree 1 coordinates: o sidering the elements by the derived brackets ( immediately that the latter are the( symmetries that preserv the canonical symmetries is hence theof algebra ( of the gauge sy JHEP01(2020)007 ) ⊕ ee . ) (3.7) ) im- E 2.15 TM ). The , Γ(
), ( α α , 2.14 3.5 ∈ ξ ) , ξ . Vielbeins ) of ( ] γ 2.14 γ = 0 G ,ξ ξ M β δ ) (as the usage ξ ∗ ξ phism between ∇ ∇ T , W, V  ( 3.1 ) G G ⊕ β M ∇ γαβ [ δ ( ∇ ioms for a generalized ∞ TM ) + ng in ( ivalent to degree 2 func- ould potentially end up + + R C γ ] ], the anchor and the pair- · d to check once more that of the induced connection, β is only responsible for the ∈ , ξ ralized vielbeins E relating , ξ he Hamiltonian for the new · ), and it is a homomorphism β βγα ) ′ , not fulfill the defining condi- .e. the formulas ( δ ξ C ) α ·i a vanishing Weitzenb¨ock-type ous symmetries, see ( ,ξ tions Γ( ∇ , ) is really a Courant algebroid E | , but it is nevertheless instruc- ( ′ in Θ for now and consider only γ h· + R [ ·i V, f ξ , Γ( G ′ , C ] ∇ α · , W ∈ h· ( i is used here, and another condition , αβγ , δ · G p ∇ ′ ) on the Courant algebroid is induced [ ] ) ) f · R β − ∇ E x , and will be chosen so that the general ξ  ( · ( ) [ = − ∇ ρ Γ( γ ,W αi E, ρ, ) ρ M ∇ V , ξ ) =

∗ α → W β α ( ξ E, ρ, ξ T ) = ρ fW α , ξ – 11 – ] E = ∇ β ∇ ⊕ γ ∇ ρ ξ ( β, γ, α Γ( =0 ∇ := ∇ G C × | V, TM ) W = ∇ Θ W E ∇ ,G , again because of associativity. In particular, ( β + ∇ β [ ) (see also point 2 above). The connection is a bona fide
∇ =0 + perm( f ξ α : Γ( C α | ∇ + , ξ ξ from generalized vector fields (which are equivalent to degr ] TM ∇ ). We will ignore the fluxes γ  V ρ ] G ξ Γ( ) γ β ξ [ 2.1 f + β ) [ →
] ∇ ) W ) to a basis for sections of ( ∇∇ α ( )-invariant pairing to the new, completely general, metric and the connection coefficients: ξ ·i E ] ρ , γ ·i [ − ∇ G h· , d, d ] , ∇ ( = ( γ : Γ( = , O ∇ β ρ [ ≡ h· β Dorf ξ [ fV ] · ) already reproduces the Courant algebroid bracket [ could have some desired features. E gives also a local isomor G ∇ ∇ , ) are still true for Θ W · [  3.7 G ∇ G by the connection of the graded symplectic manifold appeari extension to generalizedconnection: vector fields and it satisfies the ax tions), of the same symbol aims to stress), 1 functions) to ordinary tangent vector fields (which are equ where the connection via the anchor map 2.16 Before inspecting the deformation for the Courant algebroi By modifying na¨ıvely the graded symplectic structure one c M, ρ, ∗ since one can observedirect twisting that of due the bracket, to by associativity means of the tensors tensor with vari metric exact Courant algebroids, therefore mapping a basis for sec involving it is well-posed, letthe us standard comment on the presence of local gene according to definition ( tions. As already explained,deformed this setting will still not has happentive null as to Poisson long check bracket as directly with t that itself the quadruple ( describing a wanna-be Courant algebroid that actually does are sections of a frame bundle for poses a rather obvious symmetrythe condition algebraic on Bianchi the curvature identityconnection R curvature) (which is trivially true for choice ( and ( where the shorthand notation T of the bracket too. ing. Then our previous computations of the master equation i JHEP01(2020)007 ). ), M 3.1 2.8 (3.8) ) are [1] T . ) 2.16 [2] ∗ n the next E ) with van- T Γ( ignored. For to close up to 3.1 tructure ( . This general- ∈ K C U ) and ( · ifold V , · J 2.15 − ∇ , U, V ) antisymmetric Courant V given in ( onstruction in a proper f a graded Poisson struc- oth ( U M ( ad: fully generalized Dorfman ∇ bracket ion on R. Both remaining terize the structure through ∞ -preserving canonical trans- cture with the brackets ( ts of differential geometry for round fluxes e behaviour of the connection C esented here in full generality, ived the Courant algebroid on on of further correcting terms, ∈ es the failure of the antisymmet- fields ), we will start by looking for a 3.5 , f ) , V,U )-multiplication and by the antisymmetry T( = 0 M ( − is metric compatible, , in a coordinate-independent fashion. G ∞ – 12 – E ), is found to involve connec- a Weitzenb¨ock-type ∇ ∇ C ) = 3.5 U, V . For our particular way of constructing (super) gravity T( E ) gives rise to a 2-tensor if it is bilinear and antisymmetric: E , ) Γ( → U, V ) T( E . This raises some interesting points that will be explored i f Γ( M × ) = ) [1] T E arising from the derived brackets for the dg-symplectic man [2] ∗ U, fV M ), which can be interpreted as a deformation of the canonical s ∗ T T( T 3.1 ⊕ We will later apply the general formalism to the deformation To summarize, in this section we presented the general form o and the tensor T themselves under actions, we will not actuallybut need they all should the beperspective. results that interesting are in pr their own right and put the c 4.1 Generalized torsion, LieA map bracket T and : connection Γ( for such a generalbracket of Poisson this structure. algebroid, in equation The ( expression for the 4 Courant algebroid connection In this part we wouldthe like generalized to tangent formulate some bundle essential elemen section. suitable definition of a torsion tensor for ishing curvature Weitzenb¨ock-type as before and the backg TM tion for and the corresponding Hamiltonian, whichthe was master used equation. to charac formations We of then the discussed graded infinitesimal Poisson structure. degree Finally, we der simply equivalent to the statement that axioms do not require much more in order to be verified, since b which is a disguised version of the previous symmetry condit their commutator, that we could think of as a generalized Lie ized Lie bracket shall have∇ some properties determined by th of T. In previousbracket for works these there purposes, have butmostly been that because attempts choice it to led does to employ not the the obey additi a Leibniz rule. We define inste which was already required in order to produce a Poisson stru Moreover, it is a candidate forric a pair torsion of tensor, covariant if derivatives it on measur two generalized vector doing so and inspired by the structure of equation ( ture ( JHEP01(2020)007 a , (4.2) (4.1) ed Lie . Vice ), where } TM i χ ) and the pr to be also { U, V ) such that ( -holonomic) } ≡ 4.1 , E α ) dσ ρ ξ Γ( { − M or of generalized ( ) β → ξ U ∞ ) (  C E alized fashion.) ) bracket, but any two V.σ x basis, the last term in the ∈ ) (4.3) Γ( ( = 0. With the help of . (Note, that we do not − β E ⊗ K really use the generalized hor map is globally well- ) e the bracket locally. We nifold using ( ) U , f β Γ( i V K ween the infinitesimal flow y the one corresponding to l appear in the anchor. ( E ∂ nd vice versa. , ξ just as it is the case for the mmutator of vector fields — ∈ ) d that has a compatible gen- ), where it will correspond to α ry Lie bracket of vector fields x ifold. The extension can easily ξ 2 U.σ :Γ( ( J U, V i K α J , e · by definition. Its local expression ρ 1 TM f , ) · e J ized Lie bracket in terms of the canonical x Γ( + ), the standard Courant algebroid ( 4.1 i α ∈ V 2 2.5 ). We define this basis ) V to itself is , e f 1 ) − 2.7 E e ) U X,Y ˜ d ( x ∼ = h ( generalized Lie bracket ρ β + M – 13 – V ∗ K = ( i U. 2 T = 0 for a holonomic (coordinate) basis ) ∂ K ) , e ⊕ V -bilinear bracket x 1 ( Lie ( R e ] ). In fact, as one can easily check using in a holonomic basis (local Darboux chart): ρ i j α J ρ TM U, fV − ) 4.2 = , χ J ( i x V ( U, V χ K ) · α , , -section valued antisymmetric 2-tensor. Existence of such U Dorf K , is said to be a · ] U ( J E 2  ρ ) we get the following explicit expression for the generaliz , e that maps ) is recovered. V,U ≡ U, V 1 := J 4.1 e σ [ K 4.2 − -sections ) it is straightforward to obtain expressions for other (non E = An antisymmetric K U, V 4.1 The original Dorfman bracket ( J -sections U, V E J The above expression for the bracket resembles the commutat ) act on functions via the anchor map. Evaluated in a holonomic E 2 a Jacobi identity, even though it may still hold in some gener gives a local condition [ Γ( ∈ The definition does not uniquely determine a generalized Lie For what concerns us, in this article we will ultimately only A coordinate-choice independent expression for the general M 2 versa, the choice ofhave such a a natural preferred choice basisthe for can Darboux such be chart a described used preferred to before local defin equation basis, ( namel generalized Lie bracket iseralized ensured connection; for see any proposition Couranton 4.4 algebroi below. The ordina choices are related by an Lie bracket on ordinary tangent vector fields bracket, is partly given by be shown to bedefined. consistent on triplevector overlaps fields, as long in as somegenerated the by sense the anc analogously first generalized to vector the field difference on the bet second a bases. The bracket respectscan the be defining extended conditions globallytransition to functions all between patches patches of that the cover underlying the man base ma bracket of any two holonomic in the generalized sense and consequently set (generalized) solder form expression vanishes and ( corresponding expression of thebut ordinary using Lie again bracket ( as a co This expression is of course coordinate-choice dependent — the usual Lie bracket up to a rescalingExample by the 4.2. dilaton that wil Definition 4.1. U, V the defining property ( require holds for all JHEP01(2020)007 . ), i (4.8) (4.6) (4.7) (4.5) (4.4) , and U, V . It is E,E ·i . , W ) Γ( h· TM , but not E g , a connec- h∇ ∼ = pr ·i Γ( , ) ≡ exact Courant ∈ h· E ( ρ 1 Ω ⊗ ) via , W, V , ). We shall stick to the ) ) ), the fiber-wise metric ) i , E 4.6 ) M ( ( Γ( U, V , dx i V,W ∞ ; ample we cannot expect to ∇ l relativity, where the upper ∂ ; ized Lie bracket above. K indices lowered”: C t there may be several choices V,W → W U ; ( essions involving the connection ( ned on subspaces). The relevant . ∈ ) the inverse metric. This suggests . Γ Γ ) U, V ) = ( E fU i J f ( for a connection on the generalized and define connection coefficient on , θ − Γ α + i ξ Γ i V,W V, f χ = 0, and evaluated on a general section λβ U ; )) and the anchor map is g ), with all the properties required (anti- ˜ d : Γ( ), we are able to define with V W α ) = x U if for a symmetric bilinear form ξ ( E ( γα ∇ ˜ d M α V,W λ 4.1 ) = ( Γ f ∗ e h Γ( α − ∇ Γ ) V,W T ξ = (d ; f := V → ∗ ) , i.e. – 14 – := i U ρ U V ) } providing the canonical embedding of the 1-form ( U α and ( ∇ E = ∗ Γ ξ ρ fW ρ { f γαβ e Γ( is a smooth function. V,W ∇ ˜ d Γ , the following properties are required: ) = U TM ⊗ f ∇ ) = ) = ( ) h∇ of the first kind V, E U, V is trivial, i.e. ( W T( ∇ V, fW fV, W M , we will consider deformations involving a metric suitable non-degenerate symmetric bilinear form ; ; f with the appropriate tensorial and linearity properties: ∗ 5 U U T + E ( ( any ⊕ Γ Γ V -sections and ) E f is said to be ) A generalized connection symbol TM E and we employed the derivative W are ( η ρ gives and α E = ( ξ i ≡ ) TM ). The preferred basis is the one where the isomorphism of the , x h U, V, W E ( fV ( → α 1 With the help of For a generalized connection Finally, with the generalized Lie bracket ( e W E ∇ : = tion symbol of the first kind defines a generalized connection Furthermore, in section is given by the natural pairing of algebroid In the remainder of this articlecontracted we with will often a encounter expr (generalized) vector field, i.e. “with all Definition 4.3. in Ω symmetry and linearity under multiplication of functions) the generalized bundle where the same preferred basis used in the definition of the general convention for Christoffel symbolsindex of is the lowered first into kind the in last genera slot: tangent bundle a genuine torsion tensor T : Γ( with d the de Rham differential and ρ e The proof is straightforward, butof the bilinear forms important leading point to is different tha connections (some defi its inverse (theobtain inverse Levi-Civita will symbols directly, appear since theytherefore later), to do consider involve hence connection in symbols of that the first ex kind where which vanishes on the preferred basis JHEP01(2020)007 : . i s ρ 2.1 (4.9) V,V h ) ) imply . These g of the i 3 U . . ( ned when i 2 )). ρ 1 2 U, W h TM V,W = ) , U i f ] Γ( To see this we set ) ) h∇ ∈ V ( V,V TM ρ [ , but then splitting t hold, res defined before: ction of type 1. Given i Γ( ) U, X,Y first slot is obvious as can is related to axiom 2 in h nt bundle and the one on Y ) and an anchor map ∈ y clear from the context. ∀ ( ) (4.10) , . Axiom 2 and ( K E K , , s ired properties. , i.e. the axioms of a Courant ) (iv) Finally, the non-tensorial fV two brackets in their last slots , the tangent space metric can slot and vice versa. For the first V,W = 0 X s 2.1 V,V V,W ( ; J i J by s ) U h ( , and Y V ( Γ ,X,Y,Z := ) fulfills all the requirements to be a do and composition with the splitting 4.1 , s + ) ) can be replaced by TM i Γ | K X i , in the following way: TM TM | ( i s to be well-defined, the metric on the tangent Γ( , which is metric with respect to the pairing h ) and considering the properties of the pairing V,W TM ; V,W X,Y ∇ M id , i.e. metricity of the connection with respect 4.3 J → U contribute an extra term ( – 15 – ( i Z X,Y (fiberwise metric on ( h ) = K U, ∞ Γ i h h∇ s C , , because they both arise from the same connection V,V ◦ via = h h TM ∈ as in definition ) ∇ ρ i ] E K f Γ( V, fW U )) =: ), which is directly related to axiom 3 in definition J ( ), , 3 ρ Y × . J E ( 1 2 2 ) V,W with [ Γ( , s ). In order for ) ] and and pairing, U, TM → X 4.7 ) = h fU ) as in definition ( ) ∇ , s by : Γ( ); V,V are taken in the image of a generic (non-isotropic) splittin V, fW (; TM ; Z U Given a pairing ( Γ ∇ ] satisfying axioms 2 and 3 in definition Γ U s ( ( , , where an ordinary connection on vector fields is hence obtai : Γ( Γ Γ s and metricity of the connection with respect to the pairing. ) TM 2.4 2.1 , there is an interesting relation between the three structu . i . The properties to compare are the vanishing of , We shall just sketch the main points: (i) Tensoriality in the h algebroid except for the Jacobi identity, which may or may no V TM We shall denote the connection on the full generalized tange We have formulated this in the more general setting of a conne = 1. A bracket [ 3. a connection 2. a generalized Lie bracket → E tangent space by the same symbol symbols of first kind. Which one isProposition in 4.4. use should be immediatel must of course not be isotropic (because else sequence ( preserves those relations ( connection because the symbols of the first kind the arguments of to the pairing. Theterms in general the result middle follows slot by are polarization. determined by replacing then the operator be obtained from the pairing on (axiom 2) and W (iii) The antisymmetry of the generalized Lie bracket space must be non-degenerate. Using a suitable splitting case will be Via definition any two of the structures will define the third withan all ordinary its connection des cancel and imply that the connectionbracket is this tensorial in is the equivalent last to ( Proof. and the connection. (ii)exactly The mach: failures both of [ tensoriality of the be seen by replacing JHEP01(2020)007 . )  3.2 (4.12) (4.11) V,W is not ; ˜ U Γ ( are honest , section bination of Γ i f . ˜ Γ ). The first of ) ,U + ) ). It also follows i 4.10 U, V ; : necessarily have to 2.16 W, V V V,W W symmetry for torsion, h ( T( ) = h ˜ an important role here: Γ f ) β + ξ genuine generalized vector . ) and ( U i =: ) β s. We have defined a new ( . e generalized Lie bracket, the i is metric. ρ i V the r.h.s. becomes the torsion ) W . In components it reads 2.15 U , ∇ U, V ∇ . Then the connection U, V and ) = ( U, V aβγ V, Γ U h h∇ 4.2 and hence it can be labeled with the T( + T( i + ρ + = + W, i fV, W α h ) U ; ξ cβα U, V V α + ,U U here one can pick up any generalized Lie (of the first type), different than the induced Γ from the left hand side of ( ) ( i U i h∇ ˜ W, V Γ and tensorial elsewhere, hence Γ − U ; − ∇ 4.4 = – 16 – W U U V,W V K ( bγα ∇ V,W U Γ ˜ will in general have non-zero curvature. h Γ T( ) h and let ). Let us check its metricity directly: V, ∇ = ˜ f h Γ U, V ) + J ) + = C = i U ′ forms) everywhere, also in the direction along which the 4.12 ] ( − αβγ

V ′ ρ ˜ K Γ ] -linear in U V,W ) — the generalized Lie bracket is used to define torsion and R ; +( and ∇ U, V [ i U U, V U, V is always anchored with ( [ J 4.10 W, ˜ h Γ Γ − U, V ′ h = = ] is the connection induced introduced Weitzenb¨ock-type in ) i f , we obtain a connection ) V ) ) without fluxes U, V W U [ W ( follows naturally from the Jacobi identity for a certain com -dimensional index. ( V,W ρ 3.5 h d ρ ) ( W, ∇ ∇

U − ( = ) up to the generalized Lie bracket: ρ V 4.5 U ∇ Furthermore, in light of proposition Let us briefly comment at this point on the requirement of anti In conjunction with a generalized connection — that does not derivation is taken, because of the torsion tensor. derived brackets that imply the Courant algebroid axioms ( connection symbols of thefields first (i.e. both kind. vector fields Remarkably it involves since this ismetricity sometimes of omitted by other authors but it plays etebokoneWeitzenb¨ock for non-zero values of T corresponding In fact, the l.h.s. is clearly coincide with the one in ( Contracting with which is true, provided that torsion is antisymmetric and Here where the first index of anymore the one Weitzenb¨ock but has non-trivial curvature connection (of the first type) in ( from the Jacobi identity for the underlying Poisson bracket bracket, in particular the minimal one of example The antisymmetric combination oftensor covariant derivatives ( on curvature. 4.2 Connection forConsider the now derived ( deformed Courant algebroid these is matched by the onesecond following one from implies the the definition of connection th property extra terms and vice versa. JHEP01(2020)007 1 − try and ·i nian , undle (4.13) (4.14) (4.15) H TM ≡ h· is always covariant 1 G − taken with a can be shown G ), to get a bona L ∇ s naturally those 4.9 torsionless , turns out to be the ection symbols of first vative” projection onto TM | bundle, since , nd compare it with some ) ) In the pioneering work by G class of , E the full pairing . s it is the case here. But for ket): 4.5 K ], torsion is the operator T : text of curvature invariants for eneralized) vectors rather than =: )) Γ( 5 · oach will yield unphysical trivial ( h the generalized metric ∈ ! . structure, where the positive reals U, V , s as displayed in ( 1 ) TM J · ) | 1 + s ·i − . The appropriate splitting turns out X − Bg R U, , ( ; s − h· U × α, V TM | ) ); V W V i B ( d Z 1 ( ˜ ( Γ − 1 s B g O − ∇ − L ( 1 − and X,Y × − ˜ Bg Γ α V – 17 – G Z up to a rescaling by the dilaton, which is needed ) g G U ∇ V d − e 1 ∇ ( = E TM L h ∇ − | g O U G

in := W ) = = = α e ∇ · block of TM H X ) d U, V Z V e ( ∇ × . In any case, physics should not be described on the doubled T( . This formula replicates the definition of torsion in Rieman ∇ d being the adjoint representation bundle associated to the b E T 4.4 ) and hence to be a tensor like its standard differential geome F E ⊗ is the Lie derivative). With the help of the pairing, T E L 2 is compatible with a = id. Up to a similar rescaling, ), for ad ) remain true because composing with the splitting preserve ∇ s F ◦ 4.7 ρ Γ(ad ) to get some tensor field in → However, going back to the connection as defined from the conn α ) 4.8 E non-degenerate upper left to be the natural embedding of this can be achieved through a non-isotropic splitting Properties ( Coimbra, Strickland-Constable and Waldram, reference [ which is veryother similar proposals to that the have standard so definition far of appeared torsion, in a the literature. to ensure Γ( fide connection on the tangent bundle space — it just plays an auxiliary role. We do need some kind of kind, since we are interestedtheir in duals, a we connection could be that tempted returns to (g perform an inversion with of frames. T is given by the difference between a “Dorfman deri on ( This yields a bona fide connection on the generalized tangent non-degenerate if it isthe induced particular deformation by that a weresults. symplectic will consider, We structure, shall this a discuss appr thisthe point manifold, in subsection more detail in the con covariant derivative, and a “flat” one (i.e. the Dorfman brac Let us now go back to our definition of generalized torsion ( relations. 4.3 Comparison with other definitions of torsion take into account that thederivatives, frame the chosen generic is conformal. metric connection For the compatible wit geometry (where to belong to Γ(Λ In their work, counterpart. and JHEP01(2020)007 . ) 1) E − ) are 4.16 (4.16) , d 4.5 (1 ]. There, . = 0. Then . O 24 ) i 3 and using the ∇ αβγ . , e ). This 3-tensor 4.12 1 i is a section of i D e 2 E 2 α e ⊗ , e ) together with that ,W 1 D henceforth: ] E e Poisson structure has 2 3 4.5 − G ce of the bracket ( e 2 D (Λ perators for Courant al- )) from D e ot necessarily compatible al ways to construct a 3- U, V tion of generalized torsion h [ 1 ∞ be obtained by computing and from ( e l to minus that flux. This is he skew-symmetric version. e Spin bundle, indeed arise tible generalized connection, 4.8 + ut with the Dorfman bracket C also consider generalized Lie h the present one, let us show eri torsion and ours ( D − h C i d by Gualtieri in [ acket based on a Hamiltonian h ization issue of the underlying ∈ 3 ) from the Ricci tensor, which is nd acting on generalized vectors ) to define bona fide generalized ,Θ 123 , e .e. that construction is equivalent )-invariant metric (or D ] d ·i 2 Φ = 0) shall have T ( , U, V ,T ; 4.10 ∇ , e cycl h· O and eq. ( 1 , D ]: ] ]. 2 1 e W · [ ( , 25 · 26 4.3 Γ − i − 1 3 + e 2 i , e e D – 18 – Cou ] − U, W restricts to a 2 ) using [ 2 V , e e H 1 1 e e [ − ∇ h (see definition D h ) with a third generalized vector, when W, U, V V 123 ( Γ U C 4.14 ) := h∇ − cycl 2 ! ]), and refer to it as Gualtieri torsion T 1 3 , e = 7 1 ) = , e 3 ) := e 3 ) to determine a generalized Lie bracket and in ( ( , e G 2 ) that are not of Gualtieri type. W, U, V D 4.10 ( , e T 1 G 4.5 e ∇ T T( , was suggested by Alekseev and Xu in [ is compatible with the pairing. With this particular instan ·i , D h· Before elaborating on the similarities of that approach wit Another related definition, but for a covariant derivative n The connection that appears in the deformation of the graded metric in Lorentzian signature). The authors discussedgebroids. it in These the operators,naturally context which from the are of derived spin Dirac bracket2-graded connections construction. symplectic generating in manifolds The o was quant th addressed in [ with type II closed stringsbuilt bosonic with effective the commutator action of is covariantfrom derivatives recovered taking both a of the subspaces in which without flux term. With fluxa term direct the consequence of Gualtieri the torsion derivedto is bracket equa imposing construction, i the torsion condition. In fact, not identical, the latterbrackets being and more generalized general connections because nottorsions one related via can ( by ( in fact zero Gualtieri torsion with respect to the derived br is simply the contraction of ( Lie bracket to compute the torsion. The definitions of Gualti instead (as in reference [ the Courant algebroid bracketWe employed shall rather in stick the to a definition torsion is defined in t the same fashion b Moreover, it can be shownthat to we be have also given athe in special type-1 case this connection of section: symbol the defini the Gualtieri torsion can and compatible with the conformal factor Φ as well ( result both in ( a few more examplestensor out of of torsion the data tensors. givenrequiring by a in There Courant particular algebroid are that, with for other compa the natur given connection is one of the candidates for a definition “torsion” introduce Here JHEP01(2020)007 ), ⊗ , i M ture does with ∗ ): (4.17) (4.18) TM  T ,W ∗ M K Γ( ⊗ E 4.18 ∈ on 3 N M U, V g R ∗ J ⊗ defined on the = 16 Riemann T E 4 Γ( e ∇  i − h i Γ r way to ( ∈ in our case leads to re 2 ,V ontracted. ∈ -bundle. This would ,W K ′ dard one ] is article.) d differential geometry E plements the curvature ry. It also tremendously R U. X. ) are in general non-zero : Ric ) ) U, W U, V . We have also provided a R J [ . Associated with these two h extracted with the inverse of ˜ Γ ] is hence manifestly different 4.13 erse of the metric 5 + Z,Y − h W, V ( ) is the standard connection of . There are 4 of them and each i ( ) R ) R R ,U ∗ K = 4.13 = U, V E . We will eventually only make use of ; X ˜ U Γ ⊗ K ]) W ∗ ) V,W ( J E e Γ Z,Y W,V J ([ Γ( 4.12 e e ∇ ∇ i − h ) + ∈ – 19 – − − ,V ic ′ ]. The approach of [ ] U X R 5 U, W i i ; V Y V e instead, a generalized tensor in ( U, W ∇ e ∇ [ , e , Γ G Z W − e e ∇ ∇ i − h ) h h ,U ′ ] V,W ; U ( V,W [ . This procedure may seem to not make full use of the rich struc e h Γ cab = a ) = we have shown how to extract a generalized connection R ): = M 4.2 ∗ cb T W, U, V 3 ( N Ric the full generalized metric of “double geometry”, but ittensors for gives the the tangent correct bundle physics assimplifies in and the usual im Riemannian calculations. geomet not appear directly in the deformation that we consider in th The associated Ricci tensor is then the partial trace of the symmetries of the Riemann tensor can be defined in a simila Riemannian geometry. The Riemann curvature is thus the stan Since each of the indices can be vector or form valued, there a seem to lead totrivial an unphysical alternative results. way (Essentially, because to the formulate inv an action, but one consists of the sum ofIn 4 this Riemann way curvatures the with indices curvature c invariant is obtained for the full can be found again from the partial trace of this curvatures, out of which onlydefinition. one The corresponds generalized to Ricci the tensor standar G The Gualtieri torsion of the connection ( e ∇ 1. Taking into account just the tangent bundle, ( 2. Considering the connection on generalized vector fields, T as well as that of its tangent space ordinary counterpart ( the first one, but we shall briefly discuss both here. unlike the one in thefrom pioneering the work one [ presented here. 4.4 Curvature invariants In section prescription on how tooptions, evaluate it there are on also the two tangent curvature bundle invariants for only entire bundle from the derived connection of the first kind JHEP01(2020)007 , . ) ) ); ∗ 6 → ) (or M ) d, d ∗ 3.3 shall , i.e. M actor ( B g T O G [1] + ⊕ TM ∗ g . For the T ned locally ∇ : Γ( ⊕ TM n addition to , which, as an g Γ( M and are relevant and M → [1] Λ (which is in fact . separately, we can [1] G ) T H ) d ee 1 coordinates by rlying Courant alge- ( T B 2 2 M M + or ∗ Λ Λ φ and T [1] B . ill be based on ∈ ≃ in the sense that now line ate a dilaton field. Other T g and luding discussion section in the same way as ) tion on the tangent bundle. 7→ M will follow the prescriptions ion that we derive provides , can also be seen as a choice : Γ( for the connection symbols. B ensional low-energy effective ⊕ g example after equation ( E j B ( [1] M ocus on an anchor map that is ∗ at of an abelian gerbe. Such a M s of the cover of the underlying so T [1] [1] T ∗ ⊕ T ⊕ M and dilaton Γ( ): M [1] → M B T [1] ) ∗ [2] can be obtained from the standard T ∗ M T G [1] – 20 – Γ( T is invariant under coordinates. This Riemann metric ∼ = ⊕ η , 2-form is a projector up to a rescaling, the metric ) χ g and the embedding map ρ M ρ TM [1] E. The vielbein is indeed a non-canonical (differentiable ∗ η ), i.e. the data that locally defines a generalized metric and T T on the B g by applying a vielbein E based only on the Riemannian metric − : Γ( deviates from the standard pairing by an overall conformal f = E η is defined up to Λ transformations g ρ G ◦ G = B =: g T ·i ◦ ) , j B ) symmetries). This freedom means that the vielbeins are defi is trivial) and are patched globally by Λ-transformations i h· + bracket is zero, the metric d, d g dB ( p - O = -field can be introduced exploiting the structure of the unde p B . The 2-form H ) is extended to the full space of linear functions in the degr ρ ◦ M The By imposing that the anchor In the rest of the paper we will describe a particular choice f ∗ B T ◦ , in matrix notation j Γ( it is always possible to add in the vielbein an antisymmetric broid. In fact, given that the pairing part of the of section in the associated bundle of frames to and invertible) change of the degree 1 coordinates. In fact E application, is extendable to the full space also the string sigma model with background fields. Globally (unless equivalently ( and through the isomorphism Γ( choices are possible and we briefly comment on them in the conc make sure that thea relative non-trivial prefactors check of of thea the terms projection construction. in onto In the tangent this act space article up we to f a rescaling to accommod g 5.1 Deformed graded PoissonHere algebra we will present a concrete example of a deformation. It w we are dealing with a gerbe structure. By not introducing have the lower block insubsequently the the diagonal null, formulas as there discussedConsider will moreover in that the then fixand the a expression Riemannian metric composition with the anchor map 5 Deformation with metric derived Riemann tensor and subsequentlyof the the Ricci first tensor point, we namelyRic they will will be be contracted computed inaction for a for way the that closed connec will bosonic reproduce superstrings. the 10-dim constant pairing the usual transition functions.gerbe is The one overall step structure upbundles is on replace th the transition geometric functions ladder on from the a double line overlap bundle Since the JHEP01(2020)007 ) θ x ( φ (5.2) (5.4) (5.1) (5.3) and nd for χ nge of de- )) for . x a ) is hence: ( , θ , φ ) ) ( 5.2 x ( Lie f ( ] , ) hat a Riemannian λ U, U, V G i , the connection is f v ( ∂ ( v, w 1 on between G ∇ v enient to recall that 3.2 − and λ = [ = = = esponds to: ρ } } } } on . = is not globally defined, but cations for the global prop- } rdinates that defines the de- ! unction exp a are given by: v, f a B v, w v, U b dard exact Courant algebroid. U, V cal equivalence (isomorphism) { { { avity multiplet in supergravity. tz on e { , θ . . We would furthermore like to iα i will exert a force on charges. β λ δ , p i ! 0 , Γ { ∂ dB 0 1 ! 1 ba 0 1 ) − = , } )) x λ ) α ( x 1 γ x g H ( ba will in general be physically non-trivial α ( E ab a γ 2 b ): ) )) B β B E

H x i B , ξ ( 1 − λ δ 2 3.1 ∂ − i i ) λ − p B β ) ∂ γ x { ] coefficients 1 x g
( – 21 – − ( ( − 1 β g γ 28 g i ) ( λ −
E = ∂ x i 1

E η ( ∂ −

T g b λ ( E β β θ i ξ ξ ∂ + = E b . (In order to avoid confusion, let us emphasize once more = = = must be globally well-defined.) E = θ α G G iα + H λ ξ β i a , ∂ Γ χ 1 , β ) αβ j ξ − , i x λ G δ ( λ i = = 0 = = ∂ } } } } 1 is hence j j ] for a discussion in a similar setting as the one we have here a β α − , p G λ , ξ , x , ξ 27 i i i ). The connection is locally flat as it results from a local cha α p p p a ξ = { { { { and its closely associated bundles beyond the requirement t , θ } a a χ and a closed 3-form M , χ i g p { The metric The expression for E, when written by making clear distincti The local vielbein E is then completely determined, and corr ) := ( α ξ The local connection Weitzenb¨ock-type [ just like a cohomologically trivial constant magnetic field just a representative in a class locally defined by coordinates, defines the following bracket in ( a scalar field that willWith be this interpreted vielbein as the we dilaton performformation of the of the local the gr change graded of Poissonof degree structure. the 1 corresponding coo This deformed implies Courant aThis algebroid lo is to the however in stan general not a global equivalence, just lik point out that even a cohomologically trivial metric in which the conformal factor will be eventually set to some f clearly metric compatible with references to the original literature. that these are localerties considerations of and do not have any impli gree 1 coordinates. Moreover, as found in the end of subsecti The full (local) Poisson structure that arises from the ansa ( where the Poisson bracket is the canonical one, and it is conv manifold. See [ JHEP01(2020)007 i - 1 ∈ b ⊕ p − )), i oid α E dθ )) U (5.7) (5.5) (5.8) (5.6) ρ ξ ( U ∧ ρ TM ( involve  = βα ρ )( a
θ 1 )( . B M . old ) i ) − ξ B − ab G E g dp ( − B ) that appears ◦ ) for the metric Γ( ∧ β g j − ξ i ( · ∈ 3.2 ◦ v TM ab = dx α g j + ξ i , namely is invariant ( Γ( + p α η i U ) 2 b , ) ∂ i 1 ). As for the vielbein, → 2 λ ween sections of ( . e dθ ( i − of degree 1 coordinates E dx ) e substitution a v ( ∗ λ 3.1 ∧ = δ 1 ∗ 1 E  ρ − 2 , efine a generalized connection articular deformation induced − λ 1 ab ) + = λ t, as well as ( e g a overlaps of a cover of by slight abuse of notation. In . : Γ( 2 = θ ∀ = , − 1 =0 ab i U ), which is the Courant algebroid λ M − C g ′ on the element | v β  ) cancel with terms in the connection 2 . ·i 2 G ρ a [1] = 0 i ∇ v , 1 ). The new pairing (now denoted with β p ◦ − T , e } i − h· a dθ i 1 ) still holds, because it now involves the b θ ρ for this symplectic structure can be found ,
λ p ′ e 5.4 1 [2] χ 1 ] i and − ( · ( ∗ ω − θ − , c a g 1 · T 2.10 λ λ 2 [ G U i − λ dχ ∂ = ac 3 , λ – 22 – ∧ ) ). It is given by: j − , + 2 =0  E, ρ, p B λ j 2 i C  5.5 | 2 θ i − − 1 = 0 ) rather than the CCR. The direct computation results λ Θ , e g i −  dx 1 β λ a e 5.4 ρ { − h j 2 dθ ) = ( aβ λ
dx + 1 = ,U a ∧ ) looks like. We will keep identifying sections of the algebr · − ′ i i G b . This is to be compared with the approach usually taken in Gen dχ )( 2 leads to: 5.1 θ B B a ∧ , e a e θ 1  − e ⋉ a dθ ) and sections of ( ab h g ·i )) B ∧ , λ θ = j d . Elsewhere the notation is exactly as in ( i a ( h· } ∂ , M 2 3 O λ ∂ ) = ( dχ i − , e ∂ U × [1] λ + 1 3 Dorf  ) ] )( T i e · − i d { , λ ( B · dp ⊕ [ h dx i O − ∧ g − i M dx At this point we should unveil the Hamiltonian Θ The closed non-degenerate symplectic form The new pairing has a reduced symmetry group with respect to dx [1] M, ρ, = ∗ ∗ ω 5.2 Derived structure Let us describe in detailby how the the local derived vielbein structure for E this ( p in Θ can be seen to be: The inversion of the Poisson bivector agrees with this resul eralized Geometry, where the metric considered in order to d coefficients. Moreover, the composition of maps under ( compatible with the symplectic form ( Λ-transformations. This reflects again the gerbe structure a prime) is directly: deformed commutation relations ( again in and connection employed here. where the dot denotes the action of thetransitions between derivation the local Poisson structures on double T because the derivatives on the conformal factor from the canonical oneis by pull-back also with a E. homomorphism of In the fact the Poisson change algebra. Performing th The structure equation for the Hamiltonian ( in with ( with the algebra of functions of degree 1 of corresponding to the deformed Poisson algebra ( this way the vielbeinT E is also interpreted as a smooth map bet JHEP01(2020)007 : d ⊕ on − are B n (5.9) − e (5.10) (5.11) (5.12) (5.13) 2] TM g 2.1 and ). Thus ) are no , regard- λ e d : j ) ξ 5.8 [1 5.12 e + ) ( 1 ρ dλ . or ( e X ) 1 · ( )( − ρ = λ ι B 1 e factor, and the dot M. dλ . ∗ − − ) onents the very same sembles the canonical 1 2 · ) becomes:
λ ( T the Poisson bracket, is ]) + g is one of the interesting ) ρ ( 2 ) are instead: ι etween the two Courant 2] 1 )) 3.5 = d 1 e ) 2 , e N 1 − ( 2 1 5.8 e . ρ , e λ e ( urther check one could also thout 1 ) ρ roid the embedding ) here for ι e ) literature on the topic. )([ )) and ( M 2 for ( ,N behaviour w.r.t. the embedding B g − B 0 ∗ } 1 , e ) TM ) and the bracket in ( T N − 1 − 2 )) N e e + 2 g Γ( g Γ( ( ( X ( i )( g ( 5.10 2 λ ∈ .j → B and ) B , e σ ) − [1 0 1 + 2 − − ∀ e e E i N g ( h ) λX. 2 ( ρ splits into pairing, and they are neither the subbundles Dorf ) ] , X 1 , e : Γ( + ( 2 ( e + η ) 1 E ( ) = g i ρ e σ ρ – 23 – , e ( e 2] 2 h ( ( 1 L . − e ρ , e [ old λ e m + + ( j ) λ = e )) 1 [1 1 = − σ e e | ( λ ( , the axioms of the Courant algebroid definition ) in the previous subsection. Let us eliminate the metric ρ ρ i 1 E b − )( 3.2 5.7 E ) = ∈ λ ), the anchor map in ( b ) the Courant algebroid bracket in ( )), and the Leibniz rule and the Jacobi identity for ( ) does not act on the generalized vector field on which 2 B σ ) ) restricts to separate euclidean metrics of signature e ( + ). The fact that ( , e j − 5.8 5.4 ρ a )) denotes the action of the derivation by the anchored E g 5.13 E 4.15 X, σ m ( ( ( a Dorf e 1 ] ( { .j e 2 , as they were for the ) ρ H dim( ). · = ( , e h 1 2 ·i M )( ρ 1 0 , ∗ 1 h e B [ − h· T N , λ + − , which hence neglects ) invariant. Furthermore the maximal isotropic subspaces f is valued. ρ d g = = E ( ( Dorf and ′ ◦ ′ ] ] j ] · O 2 ) 2 , · TM [ B , e × , e ). 1 TM 1 ) n e − → e [ d e [ ( The definitive anchor map for the Courant algebroid is in comp This modified Dorfman bracket can be rewritten in a way that re As stressed in subsection g ( for the standard Courant algebroid: highlighted in expression ( ( ρ M, ρ, M from there and reproduce the resulting O i ∗ ∗ ◦ β old less of the particularj choice, is forced to have this scaling We can also immediately state that for an exact Courant algeb different features of this model, in contrast to theρ standard on which the metric in T For the Poisson brackets ( Their dimension is j respectively. The maximal isotropic subspaces proved for the pairingthese in ( objects yield aprove this well-defined by Courant using algebroid.algebroid the For observation brackets a that (see E f ( is a homomorphism b G longer is The parenthesis on the indices means antisymmetrization wi Here again the derivation one. To do so,also one a should homomorphism notice of that the E Dorfman being bracket a homomorphism of T JHEP01(2020)007 . ) ⊕ d ·i M 2 , h· (5.15) (5.14) , for TM are the ,..., i ). In this Dorf and using x , ] . · TM c = 1 χ δ , ˆ · χ ,M [ α -model corre- ⊗ hus one needs β σ ∧ ∗ ∧ b γ E ˆ χ and α coordinate is now M, ρ, c ∈ ∧ ∗ θ ∧ -model with ( x a T urally interpreted in h β σ ˆ ∧ χ . ⊕ α b b e, ssociated to the Poisson θ θ abc nt , . . . , d ba ∧ R βγδ ) TM b 3 plectic algebroid. lgebroid connection. We will a re in fact the point particle C = 1 e can be considered a way to e the Courant B λ θ dθ 1 6 i ymplectic structure; 6 1 mal assignation: − , ∧ + abc ∗ aph, a further application of this + g a i H E c θ . 3 Π ˆ and shifted vector fields χ + ( ab 3 λ ∧ N g θ a ∧ 1 6 Π) 2 ) becomes β χ b ∈ λ ˆ α + χ β i + C := are the 1-form associated to the degree 1 5.14 i ∧ α . x, α, b , Π ) a ( a E β θ x χ θ ∧ ( M ) maps the canonical Poisson structure to the α ∧ i ∗ ↔ bc h → i θ – 24 – a T 5.1 ) − α Q − dx ⊕ 3 γ ) canonically associated to the λ ), when the fluxes are set to zero, is that of a Courant ∧ a -model is the AKSZ action functional for a theory of 1 6 dα a σ x, ξ, p P dθ θ 5.6 ( TM ∧ + ). While doing so, the connection introduced in the de- ∧ ab β ′ c a := ˆ α B χ ·i i χ , ∂ ∧ E βγ h· 2 + b , η ′ λ θ ] a 1 2 · 1 2 , ∧ . But this comes with no surprise if one adopts the viewpoint · dχ + b [ a + i θ θ i ∧ a c ) describes a symplectic structure on the space Maps(Σ a θ -model dx dx θ ab σ E, ρ, ) as underlying symplectic algebroid is found by pullback, t ab ( ∧ ′ f ∧ 5.14 the momenta 2-forms and 3 2 i B ·i i i λ λ , i Π ∂ ) with Hamiltonian ( Π 1 2 6 1 h· 2 Σ , Σ λ ′ Z 3.1 ] + + Z · , − = · = [ i Cou ′ Cou (prior to the deformation) the projector, ( Interestingly, the momentum To conclude, let us repeat again that the derived structure a S S = Π h M, ρ, ∗ i sponding to the deformation. This As a straightforward application we can immediately comput extensively discuss it after presenting,modified in structure. the next paragr 5.2.1 Courant formed Poisson bracket setting becomes an induced Courant a extended objects with a Courant algebroid as underlying sym that the modification ofintroduce the interactions. symplectic and Poisson structur P where we have used the shorthand notation ˆ to the deformed ( algebra in ( algebroid. Since the localdeformed vielbein one, at E the in same time ( E maps the Courant algebroid ( analogue of these other ones, according to the following for coordinates. The graded variables in the previous section a The manifold Σ is the 3-dimensional worldvolume of a membran coordinates, Π By making clear distinction between shifted 1-forms Also their gradingterms and of (0-forms,) hence 1-forms their and odd/even 2-forms. parity The can deformed Coura be nat to apply the vielbein map E as T smooth dg-manifold and for sense the integral is over the 0-degree Hamiltonian for the s The functional ( JHEP01(2020)007 . l ) + α ξ 4.1 4.3 g ! and b ) 4.13 = id, α (5.18) (5.16) (5.17) a ion s δ , ξ ) =: ) ), ( ◦ a γ x ξ ρ ( . λ , 4.12 ) , dx c s ( . This will shift a E 1 ∂ ∂ − 1 Γ( λ − . Everywhere else, λ ∈ β substitute dB
U in front of the partial ca β = ved from the vielbein E erivative on generalized , which is the left inverse g ξ λ s et in the local coordinate ) H = d-result as the arguably slightly x , he connection ( ( tion for tangent space. cally zero as its definition λ X. B on of the generalized tangent ting b 1 plus the Neveu-Schwarz tensor ,U e on vector fields, instead, will ∂ ) 0 − of ves in the following way: α . a . λ c − ) G L.C. W δ ithout a rescaling by b ), is ) α ) = a 3 ξ ) = x : ′ g ( X,Y λ ) X ·i = λ ( 5.18 ( , b x g ( ∂ h· λ c − ( = 2 , s ) a ∂ ′ ,W b i β E ) δ 1 – 25 – , the particular combination of derivatives on . ) U Y − Γ( x 1) α ( , λ ! ( c 0 (1 ∂ , s λ 7→ W ) c ′ γ e c ∇ ) i + 2 ∂ ] ξ ) we will plug in the coordinate basis ( b X ( β ( δ 1 αβ s bca , ξ ) − h γ TM α x 5.12 H λ ξ ( ˜ Γ 0 0 c λ + ) respectively, where the non-isotropic splitting : Γ( a + b [ in ( dx , the connection depends only on derivatives on the conforma s ∂ ∂ γ (i.e. the connection coefficients of the first kind, see definit e 2 ( 1) bca c ∇ 1 , s U − e , e ∇ (1 dx a + L.C. λ 1 ∂ e ∇ e α a

2Γ ξ ) 3 c

)) can be collected in the following way: ∂ ) and λ W 3 a and e ( ∇ ∂ λ + ρ h ( 1) γ 4.12 , ξ = = c , s (2 ∂ -factors in the formulas, but eventually it gives the same en ′ ) i c λ = e ∇ β ∂ , ( U , ξ s 2) , α W c ξ (1 ∂ e γ ∇ To see now what is the local expression of the connection deri The components of First of all a generalized connection provides a covariant d For the components of the connection for tangent space we can ξ e , will take it away. ∇ Alternatively, one can use a projection onto tangent space w with e 3 ρ has reproduced the Christoffel symbols of first kind Γ ∇ , that here corresponds directly to the exterior derivative h β we can display thebasis, deformed i.e. Courant in algebroid place Dorfman of brack more elegant approach presented here. of The induced metric on tangent space, in light of ( derivative, carried by thenot anchor. have such factor The as covariant embedding derivativ vector fields with the split around some As expected, the differential operator comes with a factor of to the case underbundle consideration. is The briefly result discussed, as for the the main connecti focus isvector fields. on the The connec covariant derivative, by definition, beha is formulated in such frame. and equation ( In this paragraph we readily adapt the general formulas for t 5.3 Connection and curvature The same will be done for the Lie bracket, which is then identi In the first block of the first matrix , ξ is chosen to be the embedding up to the inverse of H B i.e. factor, which appears because of metricity with JHEP01(2020)007 L = .  (5.19) ik Y,W . Let us λ g = j ∂ 5.3.2 − X,V k  , ′ j and a scalar field ]) i . = is non-zero and in b  λ δ B everywhere. These i U , ∂ ), if we rather express G ca ∂ a − ) X,Y ) (5.20) e B ∇  [ ∂ g 4.2 ) and the contributions h , λ g 1 r Koszul formula to a non- b Z, λ − rule and Cartan formulas, ∂ ( c Y,X := n subsection λ X,Z ies of other non-symmetric imk ( ∂ 5.19 ( be discussed is its Gualtieri rance of a generalized Koszul and is hence invariant under : − 3 r fields ( G G g + hich is immediate if we rather H b perties of the Lie derivative − nection. ) − G c  H Z. λ δ ) mj ) g im Y.λ − as before ( x ) ( ,Y ′ ) + ( ] K 1 · B E λ − , ) = b a · λ + J ′ ∂ ), because clearly , ∂ im i Z,X X,Z  only via g a − ) ( ( B ([ 1 ∂ ) G k k Y c B − ) for the bracket, T ( 5.19 ∂ ∂ ∂ ) and λ Y. 1 − G , s − + + Y,Z ) + − λ ) ( 5.12 1 1) ik ik ) – 26 – , g 5.13 X − ) ca ) ( B b (1 λ ,X s ] B ) e m ∇ H Z,Y Z ∂ 1 + X.λ ( 1 2 ( ( s − ), for the present deformation the following connection Y,Z g ) and ( G − 1 λ ( e + ∇ ([ − h D m X. λ G given by ( mk ca ∂ 4.13 4.17  ′ b = = ] B + + ) 1 2 · i −  , ∂ · b = ( L.C. mk  , ∂ ) a mj The torsion part of the connection ( ) and ( ∂ B g c = (Γ ∂ 4.9 + X,Y ), with [ e dB. ca ∇ Z g b (  e i ∇ ˜ Γ = g ∂  4.12 2  g H . Using formula ( mj g G ), it is easy read off the connection in components: 1 2 e ∇ = 5.20 ki j In the following section we will highlight the natural appea To our knowledge, this is a totally new generalization of the ). This generalization of the Koszul formula is: e Γ x ), in equation ( ( Λ-transformations as anticipated. φ Z This expression is surelytorsion interesting. T A first thing that can We hence get the following values for the connection symbols while derivatives on the metric built up the Levi-Civitasymmetric con metric and a conformal factor. It solves ambiguit the l.h.s. before thein first the equality expression employing some ofit basic the will pro Dorfman yield bracket, the namely Koszul the formula product for an antisymmetric metric due to metricity willalso be point out analyzed that also the in connection depends a on different fashion i symbols on vector fields arise: fact equal to Employing formulas ( formula. 5.3.1 Koszul formula Let us underline here aperform peculiar property the of computation the differently. connection Focusing w on pure vecto Notice that we flipped indices picking up the opposite sign fo connection coefficients are in agreement with ( From ( JHEP01(2020)007 to r K. G (5.21) (5.22) factors. Λ reflecting d λ , + ) ]. The endomor- B M ∗ ) reminds us of the 30 T 7→ , tion for tangent vec- k ⊗ ] B ), the contorsion is the j 5.22 under consideration, we and pure torsion (where M on algebra endowed with M λg G ity condition obtained by G ] and [ ∗ i ∗ connection in the most gen- [ try c ) on a different combination T ∂ pace). Then in the derived T 29 he metric with was defined using it, because 2 1 ition, here the Koszul formula he induced metric from as in ( − n the standard GR case where 4.5 ⊗ ), and in a broader sense of the y a tensor. e ct to ∇ λ α . . α Γ(Λ 1 k M ] + − ∗ j 4.11 ∈ g k ] T 1 2 j α λg 2 ) i [ , a metric connection is obtained from + ∂ B 1 Γ(Λ ). Given that in the local coordinate basis kij + H − TM 1 λ α takes into account the variation of a metric g ∈ jki − ( g i + α [ 1 2 ∂ ) in the unique decomposition of the difference – 27 – + T ) =: kij + + ik ) we already highlighted similarities and differences H g ijk ij ) T ) = L.C. x 4.12 ( B − ∇ λ kij j + with the same torsion, see [ = ∂ kij K g Weyl term ( e E ∇ − (T k 2 1 ∂ jk g = ) := , (hence the Levi-Civita one) by adding the contorsion tenso x ( g kij 2 kij λ i T λ K ∂ ( , ) 1 is − M λ ( 2 kij 3 Ω ∈ Hence, renaming In the discussion after the definition and the derivation of a In the standard Riemannian geometry of would also like to stress that the mixed symmetry form the following compact expression is obtained The derivatives on the metric drop out as expected. K becomes directly eral setup for the derived structure ( connection with fixed torsion upon a conformal rescaling of t 1-form valued endomorphism ( phism discussed here is the vielbein E and with other works on the topic. In the specific case of the metri the torsion T of generalized vectors was noticed, and a new connection 5.3.2 Metric connectionLet with torsion us alsotors. comment briefly The on starting connection,a another which Weitzenb¨ock-type point is viewpoint of metric with on the respe discussion the was connec a graded Poiss derived brackets. between two connections for metric approaches to gravity. It enjoys also the gauge symme the difference between two metric connections is necessaril For a torsion tensor on vector fields only, T the unique metric torsion-free connection with respect to t the underlying abelian gerbethe structure. formula Moreover, is unlikepermutations derived i of from the vector three fields andis combinations assuming a a of direct torsion result the cond of metric the metric connection’s definition ( tensor K tangent space, i.e. 2 the torsion tensorbracket takes construction values the in appearance of the the generalized torsion tangent tensor ( s JHEP01(2020)007 ) .  a a ] ] d d 5.20 |  | l (5.24) (5.23) ) ) with g ). But x x ) and the H ( ( x  λ g
( 5.4 | H d ) c λ [ ∂ x l b ( g ) ∂ λ ) x ( − x tion ( ome information ( λ b L.C. a ] a λ d b ∂ | ∇ δ ∂ 1 − ) − for the connection on  − x λ a | ( ad ] c y asking for cancellation d [ g λ acd | b , whose expression ( b a 2 g n for tangent vectors were K δ t fixes all remaining ambi- e action for 10-dimensional ∂ e requirement of vanishing · R L.C. d ) , for the general connection ): )) ) , , for the deformation chosen,  sed torsion and metricity of · x φ ∇ x x sson algebra deformed with a ( J 1 e ) ( ( ∇ ic connection w.r.t. 2 λ − d λ l 4.18 ), which can in turn be obtained − l tion. In fact, it already contains λ ∂λ ∂ x ′ ∂ − ( ] ( | ·   c − L.C. [ , , the Neveu-Schwarz field ab · ) 1 ] g d ∇ d B − 2 | + (2 l λ δ ), the Riemann tensor on vector fields + − ) a ma + ] l x (2 d ( and a dilaton | 2  ) and 5.19 l we followed the prescriptions developed in H λ a − x ] a ). ( B dl H d λ | l ∂ | m ab g c – 28 – 5.3 + [ g ) b H 5.24   x | . ( ), ad 1 4 ( c H [ α g x l λ 4 1 1 2 l ( g ad −
∂ φ ) ) + . + x x da ) − l ( ( a ] x ] λ H λ d ( d H b b | (choosing the first option listed there) and presented the l λ ∂ l δ L.C. c H ∂ L.C. l ) − ∇ x 4.4 ∇ | c da 1 ( L.C. [ l c [ 2 1 ). Following therefore definition ( − λ b ∇ H a corresponds to: λ δ 1 2 + ∂ d ) , a Kalb-Ramond field c  ad 5.19 x 2 ) + ad g l − ( | ) c λ 4 [ l b L.C. 1 ∂ acd L.C. − H b ∇  ) λ ( 1 2 known as the dilaton. Prior to that, however, let us collect s − 2 − − − φ λ L.C. λ ) and the Ricci tensor Ric and subsection = (Ric R + + 4 ) and its restriction to vector fields ( ad 5.23 = ( ( One way to introduce the supergravity action functional is b To briefly recap, in this subsection Two alternative but equivalent viewpoints on the connectio Now we can proceed with computing the Riemann tensor We are now nearly ready to show the main outcome of the deforma 5.17 Ric acd ( b acd b -functions yield field equations for R e of the worldsheet conformalβ anomaly for the superstring. Th contorsion tensor given by K = on the physical model. is relevant in building theall 10-dimensional the fields supergravity of ac the supergravity multiplet, the metric the tangent bundle ( 5.4 Supergravity bosonic NS-NSIn sector this subsection we would like to show that the connection the connection, justifying the statement that it is the metr before that, let us spendclosed a strings. few words on the low energy effectiv R given. First, weresembles worked that out of a the Koszulguities standard formula in Riemannian from the geometry [ order case, of bu the entries (indices). Then we discus section explicit components expressions, inRiemannian the metric case of a graded Poi scalar field ∇ The Ricci tensor Ric JHEP01(2020)007 ]. )- ic 1) , 32 (5.26) (5.25) X,Y ) must ( just for = (1 t, it was g are taken ) = 2 [ e N ∇ 5.24 n 2 . Then one ∇ (S . dA 10) of course. E 2 , n χ = F , (10 n 2 d F =1 O X n he construction that es from the integral of 1 4 orresponding fermionic le is doubled with the i formulation, where the ). In 10 dimensions these − string effective action for ombination of symplectic the connection φ he Ricci tensors as in usual lds due to a condensate of he full dimensional spacetime over 2 on connection bined with general relativity, − 5.24 . eneralized Geometry. ity for GL(10). We then need e rangian is then valid for every ad  h Ramond boundary conditions depending on the dimensions. 2 of the sphere S IIA superstrings, with rvation: in the computations the 0) supersymmetry. The factor of ) , E Ric φ ) and ( χ ad ), comes from the symmetrization of L.C. = (2  5.23 ∇ the scalar built from Ric in ( 1 N 2 − 5.20 S -form field strengths of g n ) + 4 ( 2 B H – 29 – + 1 g 12 ( 1 − ) also the 2 − ] g ) factor. L.C.  5.25 31 B R 4  − − λ d g always occurred as direct metric while the only inverse metr B Vol ± M ) can be formulated independently from string theory: in fac . As it may not be obvious, let us remark that every 2 g Z 1 d − g 1 κ 5.25 2 2 2 − λ = ). S X,Y )( is a loop expansion parameter, and the specific factor of 2 com B To reproduce the Lagrangian in the action In either cases the geometry involved is complex geometry. T One could also possibly see this trace as in a Cartan-Palatin In 10 dimensions this action, correctly completed with the c Furthermore ( φ + 2 − g tangent. However we gave the expressionRiemannian for geometry for the the Riemann tangent and bundle, t see ( we followed here isand based on complex Generalized geometry: Geometry which they is a can c coexist if the cotangent bund employed was tangent space, rely heavily on the differential geometry of G like term, as displayed in the Koszul formula ( connection 1-form comes from the combination of a pure torsi superpartners, gives the supergravity theory for both type vielbein contributes with a ( This prescription for the traceantisymmetric relies metric in the following obse supersymmetry, and type IIB superstrings, with the worldsheet curvature, equal to the Euler character ( contain the volume form ofwhich the we (compact will subregion integrate: ofto this the) perform form 10- the is contraction naturally in a the scalar following dens way: are tensors for the general linear group GL(10) only, not of t i.e. supergravity. Seen fromdimension, this the coupling perspective, constants the getting same different lag values also developed as a field theory with local supersymmetry com e into account. From afermionic 0-mode worldsheet excitations perspective of these theon type are D-branes, II the in superstring fie short wit the R-R supersymmetric fluxes. string. This action is the low energy However it must be stressed that our prescriptions to obtain by varying the following action [ For the sake of completeness, in ( JHEP01(2020)007 s | g ), we −| (5.27) . ruction p 4.2 . 2  2 )) ) i x ) φ ( x λ ) and we con- ( L.C. λ M L.C. k ∇ ) ∇ [1] and, once projected ( T L.C. 6) + 4 ( ∇ [2] − isson structure is ready 2 ( n: ∗ d ( ld strengh, complemented T H λ 5) ( . Suitably contracted, the ) − 1 ted by an invertible change d 12 symplectic forms in a graded ach to the quest of finding a ∞ d ids with a deformed Dorfman ving the double “generalized well as the associated torsion ( curvature f type. Weitzenb¨ock Via the eralized Lie bracket ( C − λ − beins. The introduction of this cture, which can be interpreted l points of view: firstly, the con- although here we do not further h e actions as some kind of Einstein )(6 1) L.C. = 10 and , d R ) − d d −  ( d | L 3 g 2( / d ) + (1 x −| − ( φ Vol  4) p − 2 − φ . From this connection, evaluated on tangent e d M 2 H ( – 30 – Z H − λ = 1 12 il = λ x e ) gives the following lagrangian: B ) − d k 10 ( d li S 5.26 L.C. H M  R given by Z  = . The deformed graded Poisson structure already naturally . In this framework, we have presented natural construction L.C. k 4) ) is a boundary term which, for suitable boundary conditions M − M d ∗ | ∇ ∗ (10) ( ˇ g Severa and Roytenberg’s result, via a derived bracket const T 5.27 L λ T torsion | ), can be dropped. By setting g −| ⊕ 10 ⊕ x ( p −| φ Vol + p TM TM M = ∼ = Z ) ) and d ( = E x ( L d B λ (10) ) we recover the closed bosonic superstrings effective actio | , with standard ), S g x ( Our initial setting is the graded Poisson structure of Then, motivated by The explicit computation of ( −| 5.27 g TM p in ( the last parenthesis in ( tangent” bundle 6 Discussion and comments In this article wenatural have interpretation investigated of a supergravity graded andGeneral geometry string appro Relativity effectiv in a Generalized Geometry setting invol for generalized connections (ofand the curvature first tensors. and second kind) as In the corresponding action vector fields, we obtain Riemann curvature and Ricci tensors can build, out of this connection, a bona fide connection with on to involves a metric connection,derived however, bracket this construction, and connection the is introduction o of a gen we related the new graded Poissonbracket algebra on to Courant algebro by a metric andfor a quantization yielding scalar a field. kindpursue of this Thirdly, first direction). the quantized gravity deformed ( graded Po of the degree-1 coordinatesdeformed expressed graded in Poisson terms algebra of isstruction interesting local can from viel be severa seen assetting. an applications Secondly, there of Moser’s is lemma anas for underlying a bundle higher gerbe gauge stru theory with a 2-form potential and 3-form fie sider local deformations of it. The deformation is implemen JHEP01(2020)007 - H ilaton) an: the graded = 0) are already tenberg’s analysis φ he example of electro- anchor map undeformed simply the inverse of the g a role. A deformation et part, an embedding of chose the canonical one up directly related to an exact orphic to the canonical on the necessary additional y geometric structure. The ’s lemma. Note however that he graded Poisson structure, ocal coordinate changes, even amely it acts on a generalized o possible to use a construc- rmed and the Hamiltonian is iary geometric structures, but d Poisson structure in the first etic interaction and the addi- s beyond the cohomology class med graded Poisson structure: o dilaton rescaling). Here is a scalar from the tensor (involves l with respect to the Darboux , gated in a forthcoming article is ! s the NS-NS sector of supergravity 1 − ) B 1 + ). In other words the deformation can be -manifolds to Courant algebroids, one may g 2 ( compatible with the anchor map (we chose a − NQ ]. – 31 – E B 34 1 − g

] and [ ). Up to dilaton rescaling all these auxiliary geometric 33 B E = is diagonal. This feature does not undermine the validity of + G g that is just the canonical projection up to rescaling by the d s -transforms (see section ). Some partial results without dilaton (i.e. with B 5.25 ˇ Severa’s classification of exact Courant algebroids and Roy ) out of the deformed graded Poisson structure and Hamiltoni R into the generalized tangent bundle Some additional or auxiliary geometric structure is playin In this article we have focused on deformations that keep the In view of M, ( can be transformed to zero. This is not very surprising: it is 3 deformed one up to Poisson manifold and Hamiltonian thatCourant we algebroid study and are all after exact all Courant algebroids are isom local change of coordinates that areplace used and to consistency deform is the ensured grade an by a inverse graded transformation version that ofHamiltonian removes Moser the and deformation Courant of algebroidfinal t results will are deform coordinate-choice the independent. auxiliar up to rescalings. Anothera possibility deformation that based shall on be the local investi vielbein undone by a change of gradedH coordinates: since we are using l whose corresponding metric contained in our previous work [ of the correspondence of graded symplectic non-isotropic splitting in 10 dimensions ( resulting scalar curvature yields an action that reproduce tion, where the dilaton rescalingsit do will not quite appear likely in be the more auxil involved. structures are canonical and undeformed. It is probably als and finally a suitable tracethe prescription to non-symmetric obtain metric the Ricci the master equation as long as the anchor is more elaborate, n H wonder how it is possible to extract any geometric structure TM structures are also essentiallycomplete canonical list (undeformed of up thea t additional degree structures 3 beyond Hamiltonian theto that defor satisfies dilaton the rescaling), masterchart) a equation of (we generalized which Lie we bracket eventually (canonica only use the ordinary Lie brack becomes visible whenmagnetism compared in to the something introductionundeformed. the undeformed. Poisson Together, structure this In istional yields t defo structure the is correct the electromagn Hamiltonian. In the main constructi JHEP01(2020)007 ), as d ( O × ) d ( ]. However, to d exploits, the O 41 ous case. The relevance appears already directly techniques of differential theories of gravity (Gen- ramework of Generalized objects for the differential t those theories for which g f the method, let us recall nt again in the generalized II strings and the heterotic (as in string theory) and we ent vector fields, which (for he anchored vector fields is i tensor, constructed in the duced to be fore the curvature is stream- altieri torsion and taking for straints on a class of connec- orresponding deformation of ned. This fact deserves more simal symmetries on various ghtly different formulas, e.g. B es for the curvature are direct cently, e.g. in [ e deformation. This is also one in and others to find a unified ). Our construction is fully co- r supergravity from Generalized volution, while in the latter case m the deformed graded Poisson . + ) e original approaches of Einstein g ) could be very efficiently studied σ 5.6 ( 1 − (which failed). We instead encounter ) B ], and for a double field theory formula- F + 38 g ]: here the metric ( 5 − – 32 – ] and [ X 37 plays the role of a non-symmetric metric, as in the – ) = B 35 U with Hamiltonian Θ ( ( ) and the generalized torsion. We find it also interesting + ρ M g 4.2 [1] T [2] . In other parts of that work and ours there can be found quite ∗ T M ∗ as T σ ⊕ + X ]), here we present a different approach which begins with, an TM 40 = , U 39 In fact, although supergravity was already obtained in the f The approach taken here keeps the covariance of the relevant = 0) have the same expression of the eigenvectors in the previ geometry description, while atthe the generalized Lie same bracket time ( suggesting sli nonsymmetric gravitational theory investigatedfield by theory Einste involving the electromagnetic field Geometry/DFT. With this choice the computationlined of and the connection dualization and isnow there more immediate, involved. but Notice the alsousually in that expression the the literature for structure on t group the is topic of re geometric frames fo that, for the particularin ansatz his chosen, quest several to find alternativ attention. a theory The of combination gravity are consistently combi again that metricity ofoutcomes the of connection the and graded Bianchi Jacobi identiti identities. Generalized Geometry is accountablealgebra starting side. directly fro Sincethe the graded derived bracket Poisson description algebrageometry from and is a cohomology much c on lesseral doubled complicated Relativity, bundles, its than we modifications, the believefrom Supergravity this that theories algebraic setup. As an example of the convenience o our knowledge, so far there have been no attempts to construc 2-graded symplectic variant, the metric and 2-form enter in the combination tion [ do not impose any arguablytions, arbitrary but torsion rather and construct furtherof the con connection the directly striking from differences th with respect to [ vector it comes from theφ metric connection and is restricted to tang a few similarities,former for case example for in the metric thearguments generalized generalized projection vectors connection in of with the no eigenbundles the of Gu Ricc the in of the derived bracketsconfiguration in spaces of reproducing physical the interest action was pointed of out infinite re in the generalized vectormetric basis, for in a way that it is then prese Geometry as the analoguestring of (see, Einstein’s amongst gravity, the for others, the [ type JHEP01(2020)007 e ) E W, U ; ties. It V Γ( s. This may f rallelism show ) = , bilinear pairing with graded phase W W, U M theories, in the sense TM ; V ) gauge description of dealt with them in the ∇ ext is reproduced here. Poisson algebras, and as i ogical (at least off-shell), → fV f heories with various kinds U empts in this direction can cle in g. Once that this is done, el approach to gravity as a tructure. Recall that in the at in this article we neither = U, W V ear in a mutually consistent relation is given by a graded . ire a parallelizable manifold.) E ; ) gathers all the gauge fields ∇ : V W x ) = Γ( W, ρ fV h ational force is explained by means appear to be in a supersymmetric ∇ + ) + Γ( i M W, fU play a similar role as gauge potentials ; [1] V φ W, W, U T W, U ; Γ( V ), anchor V [2] V , ∇ ∗ i f M T h∇ ( – 33 – = 10 SUGRA. Another task to be accomplished + ∞ = Γ( = d i C U, W W h ) ) ∈ f f ) ) W, U h f V V ) ( ( ρ ρ V ), ) ( ρ E M fluxes: one could study them via the full Hamiltonian Θ, or ( Γ( ) = ( ) = ( -field and the dilaton γ ]. To conclude, let us mention some open questions: a complet ∞ ξ B ∈ β C 43 fW ξ ], but the other scenario opens up more intriguing possibili ( α → ξ fW, U 44 V ; ) , the , i.e. to find its geodesic equation from Hamilton’s equation : ∇ g V E αβγ ] and [ M : Γ( Γ( C U, V, W , as it should be for the abelian gerbe. Even ideas from telepa 42 [1] × H T = 0. ) } 6 [2] E ∗ Θ type 1: , T Connection Metricity Θ : Γ( The construction shares also some features with usual gauge { • • ·i , Hamiltonian, e.g. [ via the Poisson algebra. In some other works the authors have concerns the (chiral or non-chiral) representation for gauge theory elsewhere. Weof are geometry convinced can that besuch more they formulated gravity t will starting share frombe the deformed found same graded in unifying [ description. Some att in a well-defined way. We shall investigate the resulting nov would also be interesting to know the dynamics of a test parti one could attempt toderived explain from the also deformed why dg-symplectic the bosonic fields in the treatment of the R-R fields in our description is still missin for the local symmetriesgeneralization of the of dg-symplectic Moser’s manifold. lemma. The The local vielbein E( space the 3-form A Generalized differential geometry A collection ofEverywhere, fully general formulas that appeared in the t require the underlying membranei.e. sigma model to be non-topol manner and yield the bosonic part of the supergravitythat action the metric up via the connection Weitzenb¨ock inteleparallel the equivalent of deformed gravity Poisson the s sourceof of the the gravit torsion ratherneed than nor assume the a global curvature. vielbeinHere, and (Note, we all certainly however, these do th not rather requ different concepts and approaches app h· JHEP01(2020)007 ]. SPIRE ommons IN [ K ]. (1965) 286. V (1974) 498 U, V ) i J SPIRE f U 120 87 W IN ]. ( V + (1985) 159 ][ ρ ∇ V − . for support during the 54 ) i K W, SPIRE , K f W ) K IN ) arXiv:1707.00265 U V,W ][ , ( V redited. J rakidis and M. Pinkwart for U i − h U, V ( ρ rk done in collaboration with J V ρ V V,W Annals Phys. J U , − ∇ = ( = W, Extended geometry and gauged − ∇ Supergravity as Generalised K K − ∇ V V ∇ V arXiv:1107.1733 W, [ i − h h U ] W W Phys. Rev. Lett. V,W Trans. Am. Math. Soc. , ∇ U, fV J + , J i + − ∇ − ∇ U, V : ): [ U i , arXiv:1302.5419 [ K W 4.1 W U W W, V ∇ h ∇ ∇ (2011) 091 – 34 – V ∇ V,U V, V, J ∇ 11 h h ) = − ) = = ] = ) = = i : (2013) 046 ] U, V K ; JHEP V,W ), which permits any use, distribution and reproduction in , 06 U, V W V,W [ The Relativistic Spherical Top U, V ( [ T( U, V : ˜ J Γ( R W, h JHEP , : CC-BY 4.0 : i.e.: This article is distributed under the terms of the Creative C in a holonomic basis (see section 3-Cocycle in Mathematics and Physics ]. Letters to Alan WEinstein about Courant algebroids On the volume elements on a manifold ]. ]. 33 ˇ Severa, SPIRE SPIRE Generalized Lie bracket Generalized Dorfman bracket Relation between connection and brackets Torsion tensor Curvature tensor IN IN Geometry I: Type II Theories [ maximal supergravity [ • • • • • [1] R. 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